Integral of 2*sin(6x)*cos(5x) dx

1. Rewrite the integrand:

   $$2 \sin{\left(6 x \right)} \cos{\left(5 x \right)} = 1024 \sin^{5}{\left(x \right)} \cos^{6}{\left(x \right)} - 1280 \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)} + 320 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)} - 1024 \sin^{3}{\left(x \right)} \cos^{6}{\left(x \right)} + 1280 \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)} - 320 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} + 192 \sin{\left(x \right)} \cos^{6}{\left(x \right)} - 240 \sin{\left(x \right)} \cos^{4}{\left(x \right)} + 60 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$

2. Integrate term-by-term:

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

      $$\int 1024 \sin^{5}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 1024 \int \sin^{5}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

         $$\sin^{5}{\left(x \right)} \cos^{6}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$

      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 $du$:

            $$\int \left(- u^{10} + 2 u^{8} - u^{6}\right)\, du$$

            1. Integrate term-by-term:

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

                  $$\int \left(- u^{10}\right)\, du = - \int u^{10}\, du$$

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

                     $$\int u^{10}\, du = \frac{u^{11}}{11}$$

                  So, the result is: $- \frac{u^{11}}{11}$

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

                  $$\int 2 u^{8}\, du = 2 \int u^{8}\, du$$

                  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: $\frac{2 u^{9}}{9}$

               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: $- \frac{u^{7}}{7}$

               The result is: $- \frac{u^{11}}{11} + \frac{2 u^{9}}{9} - \frac{u^{7}}{7}$

            Now substitute $u$ back in:

            $$- \frac{\cos^{11}{\left(x \right)}}{11} + \frac{2 \cos^{9}{\left(x \right)}}{9} - \frac{\cos^{7}{\left(x \right)}}{7}$$

         Method #2

         1. Rewrite the integrand:

            $$\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{6}{\left(x \right)} = \sin{\left(x \right)} \cos^{10}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{8}{\left(x \right)} + \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$

         2. Integrate term-by-term:

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

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

               $$\int \left(- u^{10}\right)\, du$$

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

                  $$\int u^{10}\, du = - \int u^{10}\, du$$

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

                     $$\int u^{10}\, du = \frac{u^{11}}{11}$$

                  So, the result is: $- \frac{u^{11}}{11}$

               Now substitute $u$ back in:

               $$- \frac{\cos^{11}{\left(x \right)}}{11}$$

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

               $$\int \left(- 2 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$$

               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: $\frac{2 \cos^{9}{\left(x \right)}}{9}$

            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:

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

            The result is: $- \frac{\cos^{11}{\left(x \right)}}{11} + \frac{2 \cos^{9}{\left(x \right)}}{9} - \frac{\cos^{7}{\left(x \right)}}{7}$

         Method #3

         1. Rewrite the integrand:

            $$\left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{6}{\left(x \right)} = \sin{\left(x \right)} \cos^{10}{\left(x \right)} - 2 \sin{\left(x \right)} \cos^{8}{\left(x \right)} + \sin{\left(x \right)} \cos^{6}{\left(x \right)}$$

         2. Integrate term-by-term:

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

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

               $$\int \left(- u^{10}\right)\, du$$

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

                  $$\int u^{10}\, du = - \int u^{10}\, du$$

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

                     $$\int u^{10}\, du = \frac{u^{11}}{11}$$

                  So, the result is: $- \frac{u^{11}}{11}$

               Now substitute $u$ back in:

               $$- \frac{\cos^{11}{\left(x \right)}}{11}$$

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

               $$\int \left(- 2 \sin{\left(x \right)} \cos^{8}{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)} \cos^{8}{\left(x \right)}\, dx$$

               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: $\frac{2 \cos^{9}{\left(x \right)}}{9}$

            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:

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

            The result is: $- \frac{\cos^{11}{\left(x \right)}}{11} + \frac{2 \cos^{9}{\left(x \right)}}{9} - \frac{\cos^{7}{\left(x \right)}}{7}$

      So, the result is: $- \frac{1024 \cos^{11}{\left(x \right)}}{11} + \frac{2048 \cos^{9}{\left(x \right)}}{9} - \frac{1024 \cos^{7}{\left(x \right)}}{7}$

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

      $$\int \left(- 1280 \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 1280 \int \sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

         $$\sin^{5}{\left(x \right)} \cos^{4}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right)^{2} \sin{\left(x \right)} \cos^{4}{\left(x \right)}$$

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

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

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

         1. Integrate term-by-term:

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

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

               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: $- \frac{u^{9}}{9}$

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

               $$\int 2 u^{6}\, du = 2 \int u^{6}\, du$$

               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: $\frac{2 u^{7}}{7}$

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

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

               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: $- \frac{u^{5}}{5}$

            The result is: $- \frac{u^{9}}{9} + \frac{2 u^{7}}{7} - \frac{u^{5}}{5}$

         Now substitute $u$ back in:

         $$- \frac{\cos^{9}{\left(x \right)}}{9} + \frac{2 \cos^{7}{\left(x \right)}}{7} - \frac{\cos^{5}{\left(x \right)}}{5}$$

      So, the result is: $\frac{1280 \cos^{9}{\left(x \right)}}{9} - \frac{2560 \cos^{7}{\left(x \right)}}{7} + 256 \cos^{5}{\left(x \right)}$

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

      $$\int 320 \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 320 \int \sin^{5}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

      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: $- \frac{320 \cos^{7}{\left(x \right)}}{7} + 128 \cos^{5}{\left(x \right)} - \frac{320 \cos^{3}{\left(x \right)}}{3}$

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

      $$\int \left(- 1024 \sin^{3}{\left(x \right)} \cos^{6}{\left(x \right)}\right)\, dx = - 1024 \int \sin^{3}{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

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

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

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

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

         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:

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

               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: $- \frac{u^{7}}{7}$

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

         Now substitute $u$ back in:

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

      So, the result is: $- \frac{1024 \cos^{9}{\left(x \right)}}{9} + \frac{1024 \cos^{7}{\left(x \right)}}{7}$

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

      $$\int 1280 \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 1280 \int \sin^{3}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

      1. Rewrite the integrand:

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

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

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

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

         1. Integrate term-by-term:

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

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

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

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

               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: $- \frac{u^{5}}{5}$

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

         Now substitute $u$ back in:

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

      So, the result is: $\frac{1280 \cos^{7}{\left(x \right)}}{7} - 256 \cos^{5}{\left(x \right)}$

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

      $$\int \left(- 320 \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 320 \int \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

      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: $- 64 \cos^{5}{\left(x \right)} + \frac{320 \cos^{3}{\left(x \right)}}{3}$

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

      $$\int 192 \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx = 192 \int \sin{\left(x \right)} \cos^{6}{\left(x \right)}\, dx$$

      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: $- \frac{192 \cos^{7}{\left(x \right)}}{7}$

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

      $$\int \left(- 240 \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - 240 \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$

      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: $48 \cos^{5}{\left(x \right)}$

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

      $$\int 60 \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx = 60 \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx$$

      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: $- 20 \cos^{3}{\left(x \right)}$

   The result is: $- \frac{1024 \cos^{11}{\left(x \right)}}{11} + 256 \cos^{9}{\left(x \right)} - 256 \cos^{7}{\left(x \right)} + 112 \cos^{5}{\left(x \right)} - 20 \cos^{3}{\left(x \right)}$

3. Now simplify:

   $$\left(- \frac{1024 \cos^{8}{\left(x \right)}}{11} + 256 \cos^{6}{\left(x \right)} - 256 \cos^{4}{\left(x \right)} + 112 \cos^{2}{\left(x \right)} - 20\right) \cos^{3}{\left(x \right)}$$

4. Add the constant of integration:

   $$\left(- \frac{1024 \cos^{8}{\left(x \right)}}{11} + 256 \cos^{6}{\left(x \right)} - 256 \cos^{4}{\left(x \right)} + 112 \cos^{2}{\left(x \right)} - 20\right) \cos^{3}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\left(- \frac{1024 \cos^{8}{\left(x \right)}}{11} + 256 \cos^{6}{\left(x \right)} - 256 \cos^{4}{\left(x \right)} + 112 \cos^{2}{\left(x \right)} - 20\right) \cos^{3}{\left(x \right)}+ \mathrm{constant}$$