Integral of sin10x*sin7x dx

1. Rewrite the integrand:

   $$\sin{\left(7 x \right)} \sin{\left(10 x \right)} = - 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)} + 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)} - 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)} + 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)} - 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)} + 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 70 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

2. Integrate term-by-term:

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

      $$\int \left(- 32768 \sin^{16}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 32768 \int \sin^{16}{\left(x \right)} \cos{\left(x \right)}\, dx$$

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

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

         $$\int u^{16}\, du$$

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

            $$\int u^{16}\, du = \frac{u^{17}}{17}$$

         Now substitute $u$ back in:

         $$\frac{\sin^{17}{\left(x \right)}}{17}$$

      So, the result is: $- \frac{32768 \sin^{17}{\left(x \right)}}{17}$

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

      $$\int 122880 \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx = 122880 \int \sin^{14}{\left(x \right)} \cos{\left(x \right)}\, dx$$

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

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

         $$\int u^{14}\, du$$

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

            $$\int u^{14}\, du = \frac{u^{15}}{15}$$

         Now substitute $u$ back in:

         $$\frac{\sin^{15}{\left(x \right)}}{15}$$

      So, the result is: $8192 \sin^{15}{\left(x \right)}$

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

      $$\int \left(- 186368 \sin^{12}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 186368 \int \sin^{12}{\left(x \right)} \cos{\left(x \right)}\, dx$$

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

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

         $$\int u^{12}\, du$$

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

            $$\int u^{12}\, du = \frac{u^{13}}{13}$$

         Now substitute $u$ back in:

         $$\frac{\sin^{13}{\left(x \right)}}{13}$$

      So, the result is: $- 14336 \sin^{13}{\left(x \right)}$

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

      $$\int 146432 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx = 146432 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$$

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

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

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

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

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

         Now substitute $u$ back in:

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

      So, the result is: $13312 \sin^{11}{\left(x \right)}$

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

      $$\int \left(- 63360 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 63360 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$$

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

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

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

         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:

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

      So, the result is: $- 7040 \sin^{9}{\left(x \right)}$

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

      $$\int 14784 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 14784 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $2112 \sin^{7}{\left(x \right)}$

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

      $$\int \left(- 1680 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1680 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $- 336 \sin^{5}{\left(x \right)}$

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

      $$\int 70 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 70 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $\frac{70 \sin^{3}{\left(x \right)}}{3}$

   The result is: $- \frac{32768 \sin^{17}{\left(x \right)}}{17} + 8192 \sin^{15}{\left(x \right)} - 14336 \sin^{13}{\left(x \right)} + 13312 \sin^{11}{\left(x \right)} - 7040 \sin^{9}{\left(x \right)} + 2112 \sin^{7}{\left(x \right)} - 336 \sin^{5}{\left(x \right)} + \frac{70 \sin^{3}{\left(x \right)}}{3}$

3. Now simplify:

   $$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}$$

4. Add the constant of integration:

   $$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}$$

The answer is:

$$\frac{2 \left(- 49152 \sin^{14}{\left(x \right)} + 208896 \sin^{12}{\left(x \right)} - 365568 \sin^{10}{\left(x \right)} + 339456 \sin^{8}{\left(x \right)} - 179520 \sin^{6}{\left(x \right)} + 53856 \sin^{4}{\left(x \right)} - 8568 \sin^{2}{\left(x \right)} + 595\right) \sin^{3}{\left(x \right)}}{51}+ \mathrm{constant}$$