Integral of $$$2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}$$$

Find $$$\int \left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)d x}}} = {\color{red}{\left(- \int{2 \cos^{5}{\left(x \right)} d x} + \int{2 \cos^{8}{\left(x \right)} d x}\right)}}$$

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

$$\int{2 \cos^{8}{\left(x \right)} d x} - {\color{red}{\int{2 \cos^{5}{\left(x \right)} d x}}} = \int{2 \cos^{8}{\left(x \right)} d x} - {\color{red}{\left(2 \int{\cos^{5}{\left(x \right)} d x}\right)}}$$

Strip out one cosine and write everything else in terms of the sine, using the formula $$$\cos^2\left(\alpha \right)=-\sin^2\left(\alpha \right)+1$$$ with $$$\alpha=x$$$:

$$\int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\cos^{5}{\left(x \right)} d x}}} = \int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} d x}}}$$

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

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

The integral can be rewritten as

$$\int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} d x}}} = \int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(1 - u^{2}\right)^{2} d u}}}$$

Expand the expression:

$$\int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(1 - u^{2}\right)^{2} d u}}} = \int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(u^{4} - 2 u^{2} + 1\right)d u}}}$$

Integrate term by term:

$$\int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\int{\left(u^{4} - 2 u^{2} + 1\right)d u}}} = \int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\left(\int{1 d u} - \int{2 u^{2} d u} + \int{u^{4} d u}\right)}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$\int{2 \cos^{8}{\left(x \right)} d x} + 2 \int{2 u^{2} d u} - 2 \int{u^{4} d u} - 2 {\color{red}{\int{1 d u}}} = \int{2 \cos^{8}{\left(x \right)} d x} + 2 \int{2 u^{2} d u} - 2 \int{u^{4} d u} - 2 {\color{red}{u}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$- 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 2 \int{2 u^{2} d u} - 2 {\color{red}{\int{u^{4} d u}}}=- 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 2 \int{2 u^{2} d u} - 2 {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=- 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 2 \int{2 u^{2} d u} - 2 {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=2$$$ and $$$f{\left(u \right)} = u^{2}$$$:

$$- \frac{2 u^{5}}{5} - 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 2 {\color{red}{\int{2 u^{2} d u}}} = - \frac{2 u^{5}}{5} - 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 2 {\color{red}{\left(2 \int{u^{2} d u}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- \frac{2 u^{5}}{5} - 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 4 {\color{red}{\int{u^{2} d u}}}=- \frac{2 u^{5}}{5} - 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 4 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- \frac{2 u^{5}}{5} - 2 u + \int{2 \cos^{8}{\left(x \right)} d x} + 4 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Recall that $$$u=\sin{\left(x \right)}$$$:

$$\int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{u}} + \frac{4 {\color{red}{u}}^{3}}{3} - \frac{2 {\color{red}{u}}^{5}}{5} = \int{2 \cos^{8}{\left(x \right)} d x} - 2 {\color{red}{\sin{\left(x \right)}}} + \frac{4 {\color{red}{\sin{\left(x \right)}}}^{3}}{3} - \frac{2 {\color{red}{\sin{\left(x \right)}}}^{5}}{5}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \cos^{8}{\left(x \right)}$$$:

$$- \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + {\color{red}{\int{2 \cos^{8}{\left(x \right)} d x}}} = - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + {\color{red}{\left(2 \int{\cos^{8}{\left(x \right)} d x}\right)}}$$

Apply the power reducing formula $$$\cos^{8}{\left(\alpha \right)} = \frac{7 \cos{\left(2 \alpha \right)}}{16} + \frac{7 \cos{\left(4 \alpha \right)}}{32} + \frac{\cos{\left(6 \alpha \right)}}{16} + \frac{\cos{\left(8 \alpha \right)}}{128} + \frac{35}{128}$$$ with $$$\alpha=x$$$:

$$- \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + 2 {\color{red}{\int{\cos^{8}{\left(x \right)} d x}}} = - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + 2 {\color{red}{\int{\left(\frac{7 \cos{\left(2 x \right)}}{16} + \frac{7 \cos{\left(4 x \right)}}{32} + \frac{\cos{\left(6 x \right)}}{16} + \frac{\cos{\left(8 x \right)}}{128} + \frac{35}{128}\right)d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{128}$$$ and $$$f{\left(x \right)} = 56 \cos{\left(2 x \right)} + 28 \cos{\left(4 x \right)} + 8 \cos{\left(6 x \right)} + \cos{\left(8 x \right)} + 35$$$:

$$- \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + 2 {\color{red}{\int{\left(\frac{7 \cos{\left(2 x \right)}}{16} + \frac{7 \cos{\left(4 x \right)}}{32} + \frac{\cos{\left(6 x \right)}}{16} + \frac{\cos{\left(8 x \right)}}{128} + \frac{35}{128}\right)d x}}} = - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + 2 {\color{red}{\left(\frac{\int{\left(56 \cos{\left(2 x \right)} + 28 \cos{\left(4 x \right)} + 8 \cos{\left(6 x \right)} + \cos{\left(8 x \right)} + 35\right)d x}}{128}\right)}}$$

Integrate term by term:

$$- \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{{\color{red}{\int{\left(56 \cos{\left(2 x \right)} + 28 \cos{\left(4 x \right)} + 8 \cos{\left(6 x \right)} + \cos{\left(8 x \right)} + 35\right)d x}}}}{64} = - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{{\color{red}{\left(\int{35 d x} + \int{56 \cos{\left(2 x \right)} d x} + \int{28 \cos{\left(4 x \right)} d x} + \int{8 \cos{\left(6 x \right)} d x} + \int{\cos{\left(8 x \right)} d x}\right)}}}{64}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=35$$$:

$$- \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{8 \cos{\left(6 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{35 d x}}}}{64} = - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{8 \cos{\left(6 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\left(35 x\right)}}}{64}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=8$$$ and $$$f{\left(x \right)} = \cos{\left(6 x \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{8 \cos{\left(6 x \right)} d x}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\left(8 \int{\cos{\left(6 x \right)} d x}\right)}}}{64}$$

Let $$$u=6 x$$$.

Then $$$du=\left(6 x\right)^{\prime }dx = 6 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{6}$$$.

Therefore,

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{\cos{\left(6 x \right)} d x}}}}{8} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{6}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{6} d u}}}}{8} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{6}\right)}}}{8}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{48} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\sin{\left(u \right)}}}}{48}$$

Recall that $$$u=6 x$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{\sin{\left({\color{red}{u}} \right)}}{48} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{28 \cos{\left(4 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{\sin{\left({\color{red}{\left(6 x\right)}} \right)}}{48}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=28$$$ and $$$f{\left(x \right)} = \cos{\left(4 x \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{28 \cos{\left(4 x \right)} d x}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\left(28 \int{\cos{\left(4 x \right)} d x}\right)}}}{64}$$

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 becomes

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\cos{\left(4 x \right)} d x}}}}{16} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{16}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{16} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{16}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\sin{\left(u \right)}}}}{64}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 \sin{\left({\color{red}{u}} \right)}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{56 \cos{\left(2 x \right)} d x}}{64} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 \sin{\left({\color{red}{\left(4 x\right)}} \right)}}{64}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=56$$$ and $$$f{\left(x \right)} = \cos{\left(2 x \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\int{56 \cos{\left(2 x \right)} d x}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{{\color{red}{\left(56 \int{\cos{\left(2 x \right)} d x}\right)}}}{64}$$

Let $$$u=2 x$$$.

Then $$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{2}$$$.

Therefore,

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\cos{\left(2 x \right)} d x}}}}{8} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{8}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\frac{\cos{\left(u \right)}}{2} d u}}}}{8} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{2}\right)}}}{8}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{16} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 {\color{red}{\sin{\left(u \right)}}}}{16}$$

Recall that $$$u=2 x$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 \sin{\left({\color{red}{u}} \right)}}{16} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\int{\cos{\left(8 x \right)} d x}}{64} + \frac{7 \sin{\left({\color{red}{\left(2 x\right)}} \right)}}{16}$$

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

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

So,

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\cos{\left(8 x \right)} d x}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{64}$$

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 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{8} d u}}}}{64} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{8}\right)}}}{64}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{512} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{{\color{red}{\sin{\left(u \right)}}}}{512}$$

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

$$\frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\sin{\left({\color{red}{u}} \right)}}{512} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\sin{\left({\color{red}{\left(8 x\right)}} \right)}}{512}$$

Therefore,

$$\int{\left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)d x} = \frac{35 x}{64} - \frac{2 \sin^{5}{\left(x \right)}}{5} + \frac{4 \sin^{3}{\left(x \right)}}{3} - 2 \sin{\left(x \right)} + \frac{7 \sin{\left(2 x \right)}}{16} + \frac{7 \sin{\left(4 x \right)}}{64} + \frac{\sin{\left(6 x \right)}}{48} + \frac{\sin{\left(8 x \right)}}{512}$$

Simplify:

$$\int{\left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)d x} = \frac{4200 x - 3072 \sin^{5}{\left(x \right)} + 10240 \sin^{3}{\left(x \right)} - 15360 \sin{\left(x \right)} + 3360 \sin{\left(2 x \right)} + 840 \sin{\left(4 x \right)} + 160 \sin{\left(6 x \right)} + 15 \sin{\left(8 x \right)}}{7680}$$

Add the constant of integration:

$$\int{\left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)d x} = \frac{4200 x - 3072 \sin^{5}{\left(x \right)} + 10240 \sin^{3}{\left(x \right)} - 15360 \sin{\left(x \right)} + 3360 \sin{\left(2 x \right)} + 840 \sin{\left(4 x \right)} + 160 \sin{\left(6 x \right)} + 15 \sin{\left(8 x \right)}}{7680}+C$$

$$$\int \left(2 \cos^{8}{\left(x \right)} - 2 \cos^{5}{\left(x \right)}\right)\, dx = \frac{4200 x - 3072 \sin^{5}{\left(x \right)} + 10240 \sin^{3}{\left(x \right)} - 15360 \sin{\left(x \right)} + 3360 \sin{\left(2 x \right)} + 840 \sin{\left(4 x \right)} + 160 \sin{\left(6 x \right)} + 15 \sin{\left(8 x \right)}}{7680} + C$$$A