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

1. Rewrite the integrand:

   $$\sin{\left(5 x \right)} \sin{\left(6 x \right)} = 512 \sin^{10}{\left(x \right)} \cos{\left(x \right)} - 1152 \sin^{8}{\left(x \right)} \cos{\left(x \right)} + 896 \sin^{6}{\left(x \right)} \cos{\left(x \right)} - 280 \sin^{4}{\left(x \right)} \cos{\left(x \right)} + 30 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$

2. Integrate term-by-term:

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

      $$\int 512 \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx = 512 \int \sin^{10}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $\frac{512 \sin^{11}{\left(x \right)}}{11}$

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

      $$\int \left(- 1152 \sin^{8}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 1152 \int \sin^{8}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $- 128 \sin^{9}{\left(x \right)}$

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

      $$\int 896 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx = 896 \int \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $128 \sin^{7}{\left(x \right)}$

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

      $$\int \left(- 280 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 280 \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $- 56 \sin^{5}{\left(x \right)}$

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

      $$\int 30 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx = 30 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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: $10 \sin^{3}{\left(x \right)}$

   The result is: $\frac{512 \sin^{11}{\left(x \right)}}{11} - 128 \sin^{9}{\left(x \right)} + 128 \sin^{7}{\left(x \right)} - 56 \sin^{5}{\left(x \right)} + 10 \sin^{3}{\left(x \right)}$

3. Now simplify:

   $$\left(\frac{512 \sin^{8}{\left(x \right)}}{11} - 128 \sin^{6}{\left(x \right)} + 128 \sin^{4}{\left(x \right)} - 56 \sin^{2}{\left(x \right)} + 10\right) \sin^{3}{\left(x \right)}$$

4. Add the constant of integration:

   $$\left(\frac{512 \sin^{8}{\left(x \right)}}{11} - 128 \sin^{6}{\left(x \right)} + 128 \sin^{4}{\left(x \right)} - 56 \sin^{2}{\left(x \right)} + 10\right) \sin^{3}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\left(\frac{512 \sin^{8}{\left(x \right)}}{11} - 128 \sin^{6}{\left(x \right)} + 128 \sin^{4}{\left(x \right)} - 56 \sin^{2}{\left(x \right)} + 10\right) \sin^{3}{\left(x \right)}+ \mathrm{constant}$$