Integral of $$$\sin^{7}{\left(x \right)}$$$

Find $$$\int \sin^{7}{\left(x \right)}\, dx$$$.

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

$${\color{red}{\int{\sin^{7}{\left(x \right)} d x}}} = {\color{red}{\int{\left(1 - \cos^{2}{\left(x \right)}\right)^{3} \sin{\left(x \right)} d x}}}$$

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

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

Thus,

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

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \left(1 - u^{2}\right)^{3}$$$:

$${\color{red}{\int{\left(- \left(1 - u^{2}\right)^{3}\right)d u}}} = {\color{red}{\left(- \int{\left(1 - u^{2}\right)^{3} d u}\right)}}$$

Expand the expression:

$$- {\color{red}{\int{\left(1 - u^{2}\right)^{3} d u}}} = - {\color{red}{\int{\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\right)d u}}}$$

Integrate term by term:

$$- {\color{red}{\int{\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\right)d u}}} = - {\color{red}{\left(\int{1 d u} - \int{3 u^{2} d u} + \int{3 u^{4} d u} - \int{u^{6} d u}\right)}}$$

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

$$\int{3 u^{2} d u} - \int{3 u^{4} d u} + \int{u^{6} d u} - {\color{red}{\int{1 d u}}} = \int{3 u^{2} d u} - \int{3 u^{4} d u} + \int{u^{6} d u} - {\color{red}{u}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=6$$$:

$$- u + \int{3 u^{2} d u} - \int{3 u^{4} d u} + {\color{red}{\int{u^{6} d u}}}=- u + \int{3 u^{2} d u} - \int{3 u^{4} d u} + {\color{red}{\frac{u^{1 + 6}}{1 + 6}}}=- u + \int{3 u^{2} d u} - \int{3 u^{4} d u} + {\color{red}{\left(\frac{u^{7}}{7}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=3$$$ and $$$f{\left(u \right)} = u^{2}$$$:

$$\frac{u^{7}}{7} - u - \int{3 u^{4} d u} + {\color{red}{\int{3 u^{2} d u}}} = \frac{u^{7}}{7} - u - \int{3 u^{4} d u} + {\color{red}{\left(3 \int{u^{2} d u}\right)}}$$

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

$$\frac{u^{7}}{7} - u - \int{3 u^{4} d u} + 3 {\color{red}{\int{u^{2} d u}}}=\frac{u^{7}}{7} - u - \int{3 u^{4} d u} + 3 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=\frac{u^{7}}{7} - u - \int{3 u^{4} d u} + 3 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=3$$$ and $$$f{\left(u \right)} = u^{4}$$$:

$$\frac{u^{7}}{7} + u^{3} - u - {\color{red}{\int{3 u^{4} d u}}} = \frac{u^{7}}{7} + u^{3} - u - {\color{red}{\left(3 \int{u^{4} d u}\right)}}$$

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

$$\frac{u^{7}}{7} + u^{3} - u - 3 {\color{red}{\int{u^{4} d u}}}=\frac{u^{7}}{7} + u^{3} - u - 3 {\color{red}{\frac{u^{1 + 4}}{1 + 4}}}=\frac{u^{7}}{7} + u^{3} - u - 3 {\color{red}{\left(\frac{u^{5}}{5}\right)}}$$

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

$$- {\color{red}{u}} + {\color{red}{u}}^{3} - \frac{3 {\color{red}{u}}^{5}}{5} + \frac{{\color{red}{u}}^{7}}{7} = - {\color{red}{\cos{\left(x \right)}}} + {\color{red}{\cos{\left(x \right)}}}^{3} - \frac{3 {\color{red}{\cos{\left(x \right)}}}^{5}}{5} + \frac{{\color{red}{\cos{\left(x \right)}}}^{7}}{7}$$

Therefore,

$$\int{\sin^{7}{\left(x \right)} d x} = \frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}$$

Add the constant of integration:

$$\int{\sin^{7}{\left(x \right)} d x} = \frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}+C$$

$$$\int \sin^{7}{\left(x \right)}\, dx = \left(\frac{\cos^{7}{\left(x \right)}}{7} - \frac{3 \cos^{5}{\left(x \right)}}{5} + \cos^{3}{\left(x \right)} - \cos{\left(x \right)}\right) + C$$$A