Integral of sin(7x)*(x*cos(2x))*dx dx

1. Use integration by parts:

   $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

   Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(7 x \right)} \cos{\left(2 x \right)}$.

   Then $\operatorname{du}{\left(x \right)} = 1$.

   To find $v{\left(x \right)}$:

   1. Rewrite the integrand:

      $$\sin{\left(7 x \right)} \cos{\left(2 x \right)} = - 128 \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)} + 64 \sin^{7}{\left(x \right)} + 224 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} - 112 \sin^{5}{\left(x \right)} - 112 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} + 56 \sin^{3}{\left(x \right)} + 14 \sin{\left(x \right)} \cos^{2}{\left(x \right)} - 7 \sin{\left(x \right)}$$

   2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 128 \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 128 \int \sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{7}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

         2. There are multiple ways to do this integral.

            Method #1

            1. Let $u = \cos{\left(x \right)}$.

               Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$:

               $$\int \left(u^{8} - 3 u^{6} + 3 u^{4} - u^{2}\right)\, du$$

               1. Integrate term-by-term:

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{8}\, du = \frac{u^{9}}{9}$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int \left(- 3 u^{6}\right)\, du = - 3 \int u^{6}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{6}\, du = \frac{u^{7}}{7}$$

                     So, the result is: $- \frac{3 u^{7}}{7}$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int 3 u^{4}\, du = 3 \int u^{4}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                     So, the result is: $\frac{3 u^{5}}{5}$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                     So, the result is: $- \frac{u^{3}}{3}$

                  The result is: $\frac{u^{9}}{9} - \frac{3 u^{7}}{7} + \frac{3 u^{5}}{5} - \frac{u^{3}}{3}$

               Now substitute $u$ back in:

               $$\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$$

            Method #2

            1. Rewrite the integrand:

               $$\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{8}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

            2. Integrate term-by-term:

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{8}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{8}\, du = - \int u^{8}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{8}\, du = \frac{u^{9}}{9}$$

                        So, the result is: $- \frac{u^{9}}{9}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{9}{\left(x \right)}}{9}$$

                  So, the result is: $\frac{\cos^{9}{\left(x \right)}}{9}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{6}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{6}\, du = - \int u^{6}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{6}\, du = \frac{u^{7}}{7}$$

                        So, the result is: $- \frac{u^{7}}{7}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{7}{\left(x \right)}}{7}$$

                  So, the result is: $- \frac{3 \cos^{7}{\left(x \right)}}{7}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{4}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{4}\, du = - \int u^{4}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                        So, the result is: $- \frac{u^{5}}{5}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{5}{\left(x \right)}}{5}$$

                  So, the result is: $\frac{3 \cos^{5}{\left(x \right)}}{5}$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{2}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{2}\, du = - \int u^{2}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                     So, the result is: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

               The result is: $\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$

            Method #3

            1. Rewrite the integrand:

               $$\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{8}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

            2. Integrate term-by-term:

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{8}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{8}\, du = - \int u^{8}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{8}\, du = \frac{u^{9}}{9}$$

                        So, the result is: $- \frac{u^{9}}{9}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{9}{\left(x \right)}}{9}$$

                  So, the result is: $\frac{\cos^{9}{\left(x \right)}}{9}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int 3 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{6}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{6}\, du = - \int u^{6}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{6}\, du = \frac{u^{7}}{7}$$

                        So, the result is: $- \frac{u^{7}}{7}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{7}{\left(x \right)}}{7}$$

                  So, the result is: $- \frac{3 \cos^{7}{\left(x \right)}}{7}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

                  1. Let $u = \cos{\left(x \right)}$.

                     Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                     $$\int \left(- u^{4}\right)\, du$$

                     1. The integral of a constant times a function is the constant times the integral of the function:

                        $$\int u^{4}\, du = - \int u^{4}\, du$$

                        1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                           $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                        So, the result is: $- \frac{u^{5}}{5}$

                     Now substitute $u$ back in:

                     $$- \frac{\cos^{5}{\left(x \right)}}{5}$$

                  So, the result is: $\frac{3 \cos^{5}{\left(x \right)}}{5}$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{2}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{2}\, du = - \int u^{2}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                     So, the result is: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

               The result is: $\frac{\cos^{9}{\left(x \right)}}{9} - \frac{3 \cos^{7}{\left(x \right)}}{7} + \frac{3 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$

         So, the result is: $- \frac{128 \cos^{9}{\left(x \right)}}{9} + \frac{384 \cos^{7}{\left(x \right)}}{7} - \frac{384 \cos^{5}{\left(x \right)}}{5} + \frac{128 \cos^{3}{\left(x \right)}}{3}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 64 \sin^{7}{\left(x \right)}\, dx = 64 \int \sin^{7}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{7}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)}$$

         2. Rewrite the integrand:

            $$\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} = - \sin{\left(x \right)} \cos^{6}{\left(x \right)} + 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$$

         3. Integrate term-by-term:

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \left(- \sin{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{6}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{6}\, du = - \int u^{6}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{6}\, du = \frac{u^{7}}{7}$$

                     So, the result is: $- \frac{u^{7}}{7}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{7}{\left(x \right)}}{7}$$

               So, the result is: $\frac{\cos^{7}{\left(x \right)}}{7}$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int 3 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{4}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{4}\, du = - \int u^{4}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                     So, the result is: $- \frac{u^{5}}{5}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{5}{\left(x \right)}}{5}$$

               So, the result is: $- \frac{3 \cos^{5}{\left(x \right)}}{5}$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \left(- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 3 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{2}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{2}\, du = - \int u^{2}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                     So, the result is: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

               So, the result is: $\cos^{3}{\left(x \right)}$

            1. The integral of sine is negative cosine:

               $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

            The result is: $\frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}$

         So, the result is: $\frac{64 \cos^{7}{\left(x \right)}}{7} - \frac{192 \cos^{5}{\left(x \right)}}{5} + 64 \cos^{3}{\left(x \right)} - 64 \cos{\left(x \right)}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 224 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 224 \int \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

         2. Let $u = \cos{\left(x \right)}$.

            Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$:

            $$\int \left(- u^{6} + 2 u^{4} - u^{2}\right)\, du$$

            1. Integrate term-by-term:

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- u^{6}\right)\, du = - \int u^{6}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{6}\, du = \frac{u^{7}}{7}$$

                  So, the result is: $- \frac{u^{7}}{7}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int 2 u^{4}\, du = 2 \int u^{4}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                  So, the result is: $\frac{2 u^{5}}{5}$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                  So, the result is: $- \frac{u^{3}}{3}$

               The result is: $- \frac{u^{7}}{7} + \frac{2 u^{5}}{5} - \frac{u^{3}}{3}$

            Now substitute $u$ back in:

            $$- \frac{\cos^{7}{\left(x \right)}}{7} + \frac{2 \cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$$

         So, the result is: $- 32 \cos^{7}{\left(x \right)} + \frac{448 \cos^{5}{\left(x \right)}}{5} - \frac{224 \cos^{3}{\left(x \right)}}{3}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 112 \sin^{5}{\left(x \right)}\right)\, dx = - 112 \int \sin^{5}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{5}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)}$$

         2. Rewrite the integrand:

            $$\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} = \sin{\left(x \right)} \cos^{4}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(x \right)}$$

         3. Integrate term-by-term:

            1. Let $u = \cos{\left(x \right)}$.

               Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

               $$\int \left(- u^{4}\right)\, du$$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int u^{4}\, du = - \int u^{4}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{4}\, du = \frac{u^{5}}{5}$$

                  So, the result is: $- \frac{u^{5}}{5}$

               Now substitute $u$ back in:

               $$- \frac{\cos^{5}{\left(x \right)}}{5}$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \left(- 2 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

               1. Let $u = \cos{\left(x \right)}$.

                  Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

                  $$\int \left(- u^{2}\right)\, du$$

                  1. The integral of a constant times a function is the constant times the integral of the function:

                     $$\int u^{2}\, du = - \int u^{2}\, du$$

                     1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                        $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                     So, the result is: $- \frac{u^{3}}{3}$

                  Now substitute $u$ back in:

                  $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

               So, the result is: $\frac{2 \cos^{3}{\left(x \right)}}{3}$

            1. The integral of sine is negative cosine:

               $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

            The result is: $- \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$

         So, the result is: $\frac{112 \cos^{5}{\left(x \right)}}{5} - \frac{224 \cos^{3}{\left(x \right)}}{3} + 112 \cos{\left(x \right)}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 112 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 112 \int \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

         2. Let $u = \cos{\left(x \right)}$.

            Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$:

            $$\int \left(u^{4} - u^{2}\right)\, du$$

            1. Integrate term-by-term:

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{4}\, du = \frac{u^{5}}{5}$$

               1. The integral of a constant times a function is the constant times the integral of the function:

                  $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                  So, the result is: $- \frac{u^{3}}{3}$

               The result is: $\frac{u^{5}}{5} - \frac{u^{3}}{3}$

            Now substitute $u$ back in:

            $$\frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{112 \cos^{5}{\left(x \right)}}{5} + \frac{112 \cos^{3}{\left(x \right)}}{3}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 56 \sin^{3}{\left(x \right)}\, dx = 56 \int \sin^{3}{\left(x \right)}\, dx$$

         1. Rewrite the integrand:

            $$\sin^{3}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$

         2. Let $u = \cos{\left(x \right)}$.

            Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$:

            $$\int \left(u^{2} - 1\right)\, du$$

            1. Integrate term-by-term:

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               1. The integral of a constant is the constant times the variable of integration:

                  $$\int \left(-1\right)\, du = - u$$

               The result is: $\frac{u^{3}}{3} - u$

            Now substitute $u$ back in:

            $$\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$

         So, the result is: $\frac{56 \cos^{3}{\left(x \right)}}{3} - 56 \cos{\left(x \right)}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int 14 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 14 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

         1. Let $u = \cos{\left(x \right)}$.

            Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

            $$\int \left(- u^{2}\right)\, du$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int u^{2}\, du = - \int u^{2}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               So, the result is: $- \frac{u^{3}}{3}$

            Now substitute $u$ back in:

            $$- \frac{\cos^{3}{\left(x \right)}}{3}$$

         So, the result is: $- \frac{14 \cos^{3}{\left(x \right)}}{3}$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- 7 \sin{\left(x \right)}\right)\, dx = - 7 \int \sin{\left(x \right)}\, dx$$

         1. The integral of sine is negative cosine:

            $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

         So, the result is: $7 \cos{\left(x \right)}$

      The result is: $- \frac{128 \cos^{9}{\left(x \right)}}{9} + 32 \cos^{7}{\left(x \right)} - \frac{128 \cos^{5}{\left(x \right)}}{5} + \frac{26 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$

   Now evaluate the sub-integral.

2. Integrate term-by-term:

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- \frac{128 \cos^{9}{\left(x \right)}}{9}\right)\, dx = - \frac{128 \int \cos^{9}{\left(x \right)}\, dx}{9}$$

      1. Rewrite the integrand:

         $$\cos^{9}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{4} \cos{\left(x \right)}$$

      2. Rewrite the integrand:

         $$\left(1 - \sin^{2}{\left(x \right)}\right)^{4} \cos{\left(x \right)} = \sin^{8}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 6 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 4 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$$

      3. Integrate term-by-term:

         1. Let $u = \sin{\left(x \right)}$.

            Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

            $$\int u^{8}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{8}\, du = \frac{u^{9}}{9}$$

            Now substitute $u$ back in:

            $$\frac{\sin^{9}{\left(x \right)}}{9}$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 4 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{6}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{6}\, du = \frac{u^{7}}{7}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{7}{\left(x \right)}}{7}$$

            So, the result is: $- \frac{4 \sin^{7}{\left(x \right)}}{7}$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int 6 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 6 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{4}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{4}\, du = \frac{u^{5}}{5}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{5}{\left(x \right)}}{5}$$

            So, the result is: $\frac{6 \sin^{5}{\left(x \right)}}{5}$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 4 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 4 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{2}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{3}{\left(x \right)}}{3}$$

            So, the result is: $- \frac{4 \sin^{3}{\left(x \right)}}{3}$

         1. The integral of cosine is sine:

            $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

         The result is: $\frac{\sin^{9}{\left(x \right)}}{9} - \frac{4 \sin^{7}{\left(x \right)}}{7} + \frac{6 \sin^{5}{\left(x \right)}}{5} - \frac{4 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$

      So, the result is: $- \frac{128 \sin^{9}{\left(x \right)}}{81} + \frac{512 \sin^{7}{\left(x \right)}}{63} - \frac{256 \sin^{5}{\left(x \right)}}{15} + \frac{512 \sin^{3}{\left(x \right)}}{27} - \frac{128 \sin{\left(x \right)}}{9}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int 32 \cos^{7}{\left(x \right)}\, dx = 32 \int \cos^{7}{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

         $$\cos^{7}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{3} \cos{\left(x \right)}$$

      2. Rewrite the integrand:

         $$\left(1 - \sin^{2}{\left(x \right)}\right)^{3} \cos{\left(x \right)} = - \sin^{6}{\left(x \right)} \cos{\left(x \right)} + 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$$

      3. Integrate term-by-term:

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- \sin^{6}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{6}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{6}\, du = \frac{u^{7}}{7}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{7}{\left(x \right)}}{7}$$

            So, the result is: $- \frac{\sin^{7}{\left(x \right)}}{7}$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx = 3 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{4}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{4}\, du = \frac{u^{5}}{5}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{5}{\left(x \right)}}{5}$$

            So, the result is: $\frac{3 \sin^{5}{\left(x \right)}}{5}$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 3 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{2}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{3}{\left(x \right)}}{3}$$

            So, the result is: $- \sin^{3}{\left(x \right)}$

         1. The integral of cosine is sine:

            $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

         The result is: $- \frac{\sin^{7}{\left(x \right)}}{7} + \frac{3 \sin^{5}{\left(x \right)}}{5} - \sin^{3}{\left(x \right)} + \sin{\left(x \right)}$

      So, the result is: $- \frac{32 \sin^{7}{\left(x \right)}}{7} + \frac{96 \sin^{5}{\left(x \right)}}{5} - 32 \sin^{3}{\left(x \right)} + 32 \sin{\left(x \right)}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- \frac{128 \cos^{5}{\left(x \right)}}{5}\right)\, dx = - \frac{128 \int \cos^{5}{\left(x \right)}\, dx}{5}$$

      1. Rewrite the integrand:

         $$\cos^{5}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)}$$

      2. Rewrite the integrand:

         $$\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} = \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$$

      3. Integrate term-by-term:

         1. Let $u = \sin{\left(x \right)}$.

            Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

            $$\int u^{4}\, du$$

            1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

               $$\int u^{4}\, du = \frac{u^{5}}{5}$$

            Now substitute $u$ back in:

            $$\frac{\sin^{5}{\left(x \right)}}{5}$$

         1. The integral of a constant times a function is the constant times the integral of the function:

            $$\int \left(- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 2 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \sin{\left(x \right)}$.

               Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

               $$\int u^{2}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               Now substitute $u$ back in:

               $$\frac{\sin^{3}{\left(x \right)}}{3}$$

            So, the result is: $- \frac{2 \sin^{3}{\left(x \right)}}{3}$

         1. The integral of cosine is sine:

            $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

         The result is: $\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$

      So, the result is: $- \frac{128 \sin^{5}{\left(x \right)}}{25} + \frac{256 \sin^{3}{\left(x \right)}}{15} - \frac{128 \sin{\left(x \right)}}{5}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \frac{26 \cos^{3}{\left(x \right)}}{3}\, dx = \frac{26 \int \cos^{3}{\left(x \right)}\, dx}{3}$$

      1. Rewrite the integrand:

         $$\cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$$

      2. Let $u = \sin{\left(x \right)}$.

         Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

         $$\int \left(1 - u^{2}\right)\, du$$

         1. Integrate term-by-term:

            1. The integral of a constant is the constant times the variable of integration:

               $$\int 1\, du = u$$

            1. The integral of a constant times a function is the constant times the integral of the function:

               $$\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$$

               1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                  $$\int u^{2}\, du = \frac{u^{3}}{3}$$

               So, the result is: $- \frac{u^{3}}{3}$

            The result is: $- \frac{u^{3}}{3} + u$

         Now substitute $u$ back in:

         $$- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$$

      So, the result is: $- \frac{26 \sin^{3}{\left(x \right)}}{9} + \frac{26 \sin{\left(x \right)}}{3}$

   1. The integral of a constant times a function is the constant times the integral of the function:

      $$\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$$

      1. The integral of cosine is sine:

         $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

      So, the result is: $- \sin{\left(x \right)}$

   The result is: $- \frac{128 \sin^{9}{\left(x \right)}}{81} + \frac{32 \sin^{7}{\left(x \right)}}{9} - \frac{224 \sin^{5}{\left(x \right)}}{75} + \frac{154 \sin^{3}{\left(x \right)}}{135} - \frac{7 \sin{\left(x \right)}}{45}$

3. Now simplify:

   $$- \frac{x \left(640 \cos^{8}{\left(x \right)} - 1440 \cos^{6}{\left(x \right)} + 1152 \cos^{4}{\left(x \right)} - 390 \cos^{2}{\left(x \right)} + 45\right) \cos{\left(x \right)}}{45} + \frac{128 \sin^{9}{\left(x \right)}}{81} - \frac{32 \sin^{7}{\left(x \right)}}{9} + \frac{224 \sin^{5}{\left(x \right)}}{75} - \frac{154 \sin^{3}{\left(x \right)}}{135} + \frac{7 \sin{\left(x \right)}}{45}$$

4. Add the constant of integration:

   $$- \frac{x \left(640 \cos^{8}{\left(x \right)} - 1440 \cos^{6}{\left(x \right)} + 1152 \cos^{4}{\left(x \right)} - 390 \cos^{2}{\left(x \right)} + 45\right) \cos{\left(x \right)}}{45} + \frac{128 \sin^{9}{\left(x \right)}}{81} - \frac{32 \sin^{7}{\left(x \right)}}{9} + \frac{224 \sin^{5}{\left(x \right)}}{75} - \frac{154 \sin^{3}{\left(x \right)}}{135} + \frac{7 \sin{\left(x \right)}}{45}+ \mathrm{constant}$$

The answer is:

$$- \frac{x \left(640 \cos^{8}{\left(x \right)} - 1440 \cos^{6}{\left(x \right)} + 1152 \cos^{4}{\left(x \right)} - 390 \cos^{2}{\left(x \right)} + 45\right) \cos{\left(x \right)}}{45} + \frac{128 \sin^{9}{\left(x \right)}}{81} - \frac{32 \sin^{7}{\left(x \right)}}{9} + \frac{224 \sin^{5}{\left(x \right)}}{75} - \frac{154 \sin^{3}{\left(x \right)}}{135} + \frac{7 \sin{\left(x \right)}}{45}+ \mathrm{constant}$$