Integral of $$$\sin^{2}{\left(x \right)} \cos^{3}{\left(3 x \right)}$$$

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

Apply the power reducing formula $$$\sin^{2}{\left(\alpha \right)} = \frac{1}{2} - \frac{\cos{\left(2 \alpha \right)}}{2}$$$ with $$$\alpha=x$$$:

$${\color{red}{\int{\sin^{2}{\left(x \right)} \cos^{3}{\left(3 x \right)} d x}}} = {\color{red}{\int{\frac{\left(1 - \cos{\left(2 x \right)}\right) \cos^{3}{\left(3 x \right)}}{2} d x}}}$$

Apply the power reducing formula $$$\cos^{3}{\left(\alpha \right)} = \frac{3 \cos{\left(\alpha \right)}}{4} + \frac{\cos{\left(3 \alpha \right)}}{4}$$$ with $$$\alpha=3 x$$$:

$${\color{red}{\int{\frac{\left(1 - \cos{\left(2 x \right)}\right) \cos^{3}{\left(3 x \right)}}{2} d x}}} = {\color{red}{\int{\frac{\left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right)}{8} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{8}$$$ and $$$f{\left(x \right)} = \left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right)$$$:

$${\color{red}{\int{\frac{\left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right)}{8} d x}}} = {\color{red}{\left(\frac{\int{\left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right) d x}}{8}\right)}}$$

Expand the expression:

$$\frac{{\color{red}{\int{\left(1 - \cos{\left(2 x \right)}\right) \left(3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right) d x}}}}{8} = \frac{{\color{red}{\int{\left(- 3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} - \cos{\left(2 x \right)} \cos{\left(9 x \right)} + 3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right)d x}}}}{8}$$

Integrate term by term:

$$\frac{{\color{red}{\int{\left(- 3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} - \cos{\left(2 x \right)} \cos{\left(9 x \right)} + 3 \cos{\left(3 x \right)} + \cos{\left(9 x \right)}\right)d x}}}}{8} = \frac{{\color{red}{\left(- \int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x} - \int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x} + \int{3 \cos{\left(3 x \right)} d x} + \int{\cos{\left(9 x \right)} d x}\right)}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$:

$$- \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{{\color{red}{\int{3 \cos{\left(3 x \right)} d x}}}}{8} = - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{{\color{red}{\left(3 \int{\cos{\left(3 x \right)} d x}\right)}}}{8}$$

Let $$$u=3 x$$$.

Then $$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{3}$$$.

So,

$$- \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{3 {\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{8} = - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{8} = - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{3}\right)}}}{8}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{{\color{red}{\sin{\left(u \right)}}}}{8}$$

Recall that $$$u=3 x$$$:

$$- \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{u}} \right)}}{8} = - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} - \frac{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} + \frac{\sin{\left({\color{red}{\left(3 x\right)}} \right)}}{8}$$

Rewrite the integrand using the formula $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ with $$$\alpha=2 x$$$ and $$$\beta=9 x$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(2 x \right)} \cos{\left(9 x \right)} d x}}}}{8} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(7 x \right)}}{2} + \frac{\cos{\left(11 x \right)}}{2}\right)d x}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = \cos{\left(7 x \right)} + \cos{\left(11 x \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{\cos{\left(7 x \right)}}{2} + \frac{\cos{\left(11 x \right)}}{2}\right)d x}}}}{8} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(\cos{\left(7 x \right)} + \cos{\left(11 x \right)}\right)d x}}{2}\right)}}}{8}$$

Integrate term by term:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\cos{\left(7 x \right)} + \cos{\left(11 x \right)}\right)d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(\int{\cos{\left(7 x \right)} d x} + \int{\cos{\left(11 x \right)} d x}\right)}}}{16}$$

Let $$$u=7 x$$$.

Then $$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{7}$$$.

Thus,

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{16}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{7}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}}{16}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{112} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{{\color{red}{\sin{\left(u \right)}}}}{112}$$

Recall that $$$u=7 x$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{\sin{\left({\color{red}{u}} \right)}}{112} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\int{\cos{\left(11 x \right)} d x}}{16} - \frac{\sin{\left({\color{red}{\left(7 x\right)}} \right)}}{112}$$

Let $$$u=11 x$$$.

Then $$$du=\left(11 x\right)^{\prime }dx = 11 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{11}$$$.

Therefore,

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(11 x \right)} d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{11} d u}}}}{16}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{11}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{11} d u}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{11}\right)}}}{16}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{176} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\sin{\left(u \right)}}}}{176}$$

Recall that $$$u=11 x$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\sin{\left({\color{red}{u}} \right)}}{176} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}{8} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{\sin{\left({\color{red}{\left(11 x\right)}} \right)}}{176}$$

Rewrite $$$\cos\left(2 x \right)\cos\left(3 x \right)$$$ using the formula $$$\cos\left(\alpha \right)\cos\left(\beta \right)=\frac{1}{2} \cos\left(\alpha-\beta \right)+\frac{1}{2} \cos\left(\alpha+\beta \right)$$$ with $$$\alpha=2 x$$$ and $$$\beta=3 x$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{3 \cos{\left(2 x \right)} \cos{\left(3 x \right)} d x}}}}{8} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(5 x \right)}}{2}\right)d x}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = 3 \cos{\left(x \right)} + 3 \cos{\left(5 x \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(\frac{3 \cos{\left(x \right)}}{2} + \frac{3 \cos{\left(5 x \right)}}{2}\right)d x}}}}{8} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(\frac{\int{\left(3 \cos{\left(x \right)} + 3 \cos{\left(5 x \right)}\right)d x}}{2}\right)}}}{8}$$

Integrate term by term:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{\left(3 \cos{\left(x \right)} + 3 \cos{\left(5 x \right)}\right)d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(\int{3 \cos{\left(x \right)} d x} + \int{3 \cos{\left(5 x \right)} d x}\right)}}}{16}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} - \frac{\int{3 \cos{\left(5 x \right)} d x}}{16} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{3 \cos{\left(x \right)} d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} - \frac{\int{3 \cos{\left(5 x \right)} d x}}{16} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(3 \int{\cos{\left(x \right)} d x}\right)}}}{16}$$

The integral of the cosine is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$\frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} - \frac{\int{3 \cos{\left(5 x \right)} d x}}{16} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\cos{\left(x \right)} d x}}}}{16} = \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} - \frac{\int{3 \cos{\left(5 x \right)} d x}}{16} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\sin{\left(x \right)}}}}{16}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=3$$$ and $$$f{\left(x \right)} = \cos{\left(5 x \right)}$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\int{3 \cos{\left(5 x \right)} d x}}}}{16} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{{\color{red}{\left(3 \int{\cos{\left(5 x \right)} d x}\right)}}}{16}$$

Let $$$u=5 x$$$.

Then $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{5}$$$.

The integral becomes

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\cos{\left(5 x \right)} d x}}}}{16} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{5} d u}}}}{16}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\frac{\cos{\left(u \right)}}{5} d u}}}}{16} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{5}\right)}}}{16}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{80} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 {\color{red}{\sin{\left(u \right)}}}}{80}$$

Recall that $$$u=5 x$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 \sin{\left({\color{red}{u}} \right)}}{80} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\int{\cos{\left(9 x \right)} d x}}{8} - \frac{3 \sin{\left({\color{red}{\left(5 x\right)}} \right)}}{80}$$

Let $$$u=9 x$$$.

Then $$$du=\left(9 x\right)^{\prime }dx = 9 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{9}$$$.

Thus,

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\int{\cos{\left(9 x \right)} d x}}}}{8} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{9} d u}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{9}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{9} d u}}}}{8} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{9}\right)}}}{8}$$

The integral of the cosine is $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{72} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{{\color{red}{\sin{\left(u \right)}}}}{72}$$

Recall that $$$u=9 x$$$:

$$- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\sin{\left({\color{red}{u}} \right)}}{72} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} - \frac{\sin{\left(11 x \right)}}{176} + \frac{\sin{\left({\color{red}{\left(9 x\right)}} \right)}}{72}$$

Therefore,

$$\int{\sin^{2}{\left(x \right)} \cos^{3}{\left(3 x \right)} d x} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} + \frac{\sin{\left(9 x \right)}}{72} - \frac{\sin{\left(11 x \right)}}{176}$$

Add the constant of integration:

$$\int{\sin^{2}{\left(x \right)} \cos^{3}{\left(3 x \right)} d x} = - \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} + \frac{\sin{\left(9 x \right)}}{72} - \frac{\sin{\left(11 x \right)}}{176}+C$$

$$$\int \sin^{2}{\left(x \right)} \cos^{3}{\left(3 x \right)}\, dx = \left(- \frac{3 \sin{\left(x \right)}}{16} + \frac{\sin{\left(3 x \right)}}{8} - \frac{3 \sin{\left(5 x \right)}}{80} - \frac{\sin{\left(7 x \right)}}{112} + \frac{\sin{\left(9 x \right)}}{72} - \frac{\sin{\left(11 x \right)}}{176}\right) + C$$$A