Integral of $$$\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}$$$

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

Rewrite the integrand using the double angle formula $$$\sin\left(x \right)\cos\left(x \right)=\frac{1}{2}\sin\left( 2 x \right)$$$:

$${\color{red}{\int{\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)} d x}}} = {\color{red}{\int{\frac{\sin^{4}{\left(2 x \right)}}{16} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{16}$$$ and $$$f{\left(x \right)} = \sin^{4}{\left(2 x \right)}$$$:

$${\color{red}{\int{\frac{\sin^{4}{\left(2 x \right)}}{16} d x}}} = {\color{red}{\left(\frac{\int{\sin^{4}{\left(2 x \right)} d x}}{16}\right)}}$$

Apply the power reducing formula $$$\sin^{4}{\left(\alpha \right)} = - \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$ with $$$\alpha=2 x$$$:

$$\frac{{\color{red}{\int{\sin^{4}{\left(2 x \right)} d x}}}}{16} = \frac{{\color{red}{\int{\left(- \frac{\cos{\left(4 x \right)}}{2} + \frac{\cos{\left(8 x \right)}}{8} + \frac{3}{8}\right)d x}}}}{16}$$

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 $$$f{\left(x \right)} = - 4 \cos{\left(4 x \right)} + \cos{\left(8 x \right)} + 3$$$:

$$\frac{{\color{red}{\int{\left(- \frac{\cos{\left(4 x \right)}}{2} + \frac{\cos{\left(8 x \right)}}{8} + \frac{3}{8}\right)d x}}}}{16} = \frac{{\color{red}{\left(\frac{\int{\left(- 4 \cos{\left(4 x \right)} + \cos{\left(8 x \right)} + 3\right)d x}}{8}\right)}}}{16}$$

Integrate term by term:

$$\frac{{\color{red}{\int{\left(- 4 \cos{\left(4 x \right)} + \cos{\left(8 x \right)} + 3\right)d x}}}}{128} = \frac{{\color{red}{\left(\int{3 d x} - \int{4 \cos{\left(4 x \right)} d x} + \int{\cos{\left(8 x \right)} d x}\right)}}}{128}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=3$$$:

$$- \frac{\int{4 \cos{\left(4 x \right)} d x}}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} + \frac{{\color{red}{\int{3 d x}}}}{128} = - \frac{\int{4 \cos{\left(4 x \right)} d x}}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} + \frac{{\color{red}{\left(3 x\right)}}}{128}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$:

$$\frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\int{4 \cos{\left(4 x \right)} d x}}}}{128} = \frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\left(4 \int{\cos{\left(4 x \right)} d x}\right)}}}{128}$$

Let $$$u=4 x$$$.

Then $$$du=\left(4 x\right)^{\prime }dx = 4 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{4}$$$.

The integral can be rewritten as

$$\frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{32} = \frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{32}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{32} = \frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{32}$$

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

$$\frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{128} = \frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{{\color{red}{\sin{\left(u \right)}}}}{128}$$

Recall that $$$u=4 x$$$:

$$\frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{\sin{\left({\color{red}{u}} \right)}}{128} = \frac{3 x}{128} + \frac{\int{\cos{\left(8 x \right)} d x}}{128} - \frac{\sin{\left({\color{red}{\left(4 x\right)}} \right)}}{128}$$

Let $$$u=8 x$$$.

Then $$$du=\left(8 x\right)^{\prime }dx = 8 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{8}$$$.

Thus,

$$\frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\cos{\left(8 x \right)} d x}}}}{128} = \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{128}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{8}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{128} = \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}}{128}$$

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

$$\frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{1024} = \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{{\color{red}{\sin{\left(u \right)}}}}{1024}$$

Recall that $$$u=8 x$$$:

$$\frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{\sin{\left({\color{red}{u}} \right)}}{1024} = \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{\sin{\left({\color{red}{\left(8 x\right)}} \right)}}{1024}$$

Therefore,

$$\int{\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)} d x} = \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{\sin{\left(8 x \right)}}{1024}$$

Simplify:

$$\int{\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)} d x} = \frac{24 x - 8 \sin{\left(4 x \right)} + \sin{\left(8 x \right)}}{1024}$$

Add the constant of integration:

$$\int{\sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)} d x} = \frac{24 x - 8 \sin{\left(4 x \right)} + \sin{\left(8 x \right)}}{1024}+C$$

$$$\int \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = \frac{24 x - 8 \sin{\left(4 x \right)} + \sin{\left(8 x \right)}}{1024} + C$$$A