Integral of $$$\frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}}$$$

Find $$$\int \frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}}\, dx$$$.

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 $$$\alpha=x$$$:

$${\color{red}{\int{\frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x}}} = {\color{red}{\int{\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{4} \sin{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x}}}$$

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 $$$\sin{\left(x \right)} dx = - du$$$.

Therefore,

$${\color{red}{\int{\frac{\left(1 - \cos^{2}{\left(x \right)}\right)^{4} \sin{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x}}} = {\color{red}{\int{\left(- \frac{\left(1 - u^{2}\right)^{4}}{u^{11}}\right)d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=-1$$$ and $$$f{\left(u \right)} = \frac{\left(1 - u^{2}\right)^{4}}{u^{11}}$$$:

$${\color{red}{\int{\left(- \frac{\left(1 - u^{2}\right)^{4}}{u^{11}}\right)d u}}} = {\color{red}{\left(- \int{\frac{\left(1 - u^{2}\right)^{4}}{u^{11}} d u}\right)}}$$

Expand the expression:

$$- {\color{red}{\int{\frac{\left(1 - u^{2}\right)^{4}}{u^{11}} d u}}} = - {\color{red}{\int{\left(\frac{1}{u^{3}} - \frac{4}{u^{5}} + \frac{6}{u^{7}} - \frac{4}{u^{9}} + \frac{1}{u^{11}}\right)d u}}}$$

Integrate term by term:

$$- {\color{red}{\int{\left(\frac{1}{u^{3}} - \frac{4}{u^{5}} + \frac{6}{u^{7}} - \frac{4}{u^{9}} + \frac{1}{u^{11}}\right)d u}}} = - {\color{red}{\left(\int{\frac{1}{u^{11}} d u} - \int{\frac{4}{u^{9}} d u} + \int{\frac{6}{u^{7}} d u} - \int{\frac{4}{u^{5}} d u} + \int{\frac{1}{u^{3}} d u}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-11$$$:

$$\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - \int{\frac{1}{u^{3}} d u} - {\color{red}{\int{\frac{1}{u^{11}} d u}}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - \int{\frac{1}{u^{3}} d u} - {\color{red}{\int{u^{-11} d u}}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - \int{\frac{1}{u^{3}} d u} - {\color{red}{\frac{u^{-11 + 1}}{-11 + 1}}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - \int{\frac{1}{u^{3}} d u} - {\color{red}{\left(- \frac{u^{-10}}{10}\right)}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - \int{\frac{1}{u^{3}} d u} - {\color{red}{\left(- \frac{1}{10 u^{10}}\right)}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-3$$$:

$$\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - {\color{red}{\int{\frac{1}{u^{3}} d u}}} + \frac{1}{10 u^{10}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - {\color{red}{\int{u^{-3} d u}}} + \frac{1}{10 u^{10}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - {\color{red}{\frac{u^{-3 + 1}}{-3 + 1}}} + \frac{1}{10 u^{10}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - {\color{red}{\left(- \frac{u^{-2}}{2}\right)}} + \frac{1}{10 u^{10}}=\int{\frac{4}{u^{9}} d u} - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} - {\color{red}{\left(- \frac{1}{2 u^{2}}\right)}} + \frac{1}{10 u^{10}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=4$$$ and $$$f{\left(u \right)} = \frac{1}{u^{9}}$$$:

$$- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + {\color{red}{\int{\frac{4}{u^{9}} d u}}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}} = - \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + {\color{red}{\left(4 \int{\frac{1}{u^{9}} d u}\right)}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-9$$$:

$$- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + 4 {\color{red}{\int{\frac{1}{u^{9}} d u}}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + 4 {\color{red}{\int{u^{-9} d u}}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + 4 {\color{red}{\frac{u^{-9 + 1}}{-9 + 1}}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + 4 {\color{red}{\left(- \frac{u^{-8}}{8}\right)}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + \int{\frac{4}{u^{5}} d u} + 4 {\color{red}{\left(- \frac{1}{8 u^{8}}\right)}} + \frac{1}{2 u^{2}} + \frac{1}{10 u^{10}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=4$$$ and $$$f{\left(u \right)} = \frac{1}{u^{5}}$$$:

$$- \int{\frac{6}{u^{7}} d u} + {\color{red}{\int{\frac{4}{u^{5}} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}} = - \int{\frac{6}{u^{7}} d u} + {\color{red}{\left(4 \int{\frac{1}{u^{5}} d u}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-5$$$:

$$- \int{\frac{6}{u^{7}} d u} + 4 {\color{red}{\int{\frac{1}{u^{5}} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + 4 {\color{red}{\int{u^{-5} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + 4 {\color{red}{\frac{u^{-5 + 1}}{-5 + 1}}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + 4 {\color{red}{\left(- \frac{u^{-4}}{4}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- \int{\frac{6}{u^{7}} d u} + 4 {\color{red}{\left(- \frac{1}{4 u^{4}}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=6$$$ and $$$f{\left(u \right)} = \frac{1}{u^{7}}$$$:

$$- {\color{red}{\int{\frac{6}{u^{7}} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}} = - {\color{red}{\left(6 \int{\frac{1}{u^{7}} d u}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}$$

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-7$$$:

$$- 6 {\color{red}{\int{\frac{1}{u^{7}} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- 6 {\color{red}{\int{u^{-7} d u}}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- 6 {\color{red}{\frac{u^{-7 + 1}}{-7 + 1}}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- 6 {\color{red}{\left(- \frac{u^{-6}}{6}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}=- 6 {\color{red}{\left(- \frac{1}{6 u^{6}}\right)}} + \frac{1}{2 u^{2}} - \frac{1}{u^{4}} - \frac{1}{2 u^{8}} + \frac{1}{10 u^{10}}$$

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

$$\frac{{\color{red}{u}}^{-10}}{10} - \frac{{\color{red}{u}}^{-8}}{2} + {\color{red}{u}}^{-6} - {\color{red}{u}}^{-4} + \frac{{\color{red}{u}}^{-2}}{2} = \frac{{\color{red}{\cos{\left(x \right)}}}^{-10}}{10} - \frac{{\color{red}{\cos{\left(x \right)}}}^{-8}}{2} + {\color{red}{\cos{\left(x \right)}}}^{-6} - {\color{red}{\cos{\left(x \right)}}}^{-4} + \frac{{\color{red}{\cos{\left(x \right)}}}^{-2}}{2}$$

Therefore,

$$\int{\frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x} = \frac{1}{2 \cos^{2}{\left(x \right)}} - \frac{1}{\cos^{4}{\left(x \right)}} + \frac{1}{\cos^{6}{\left(x \right)}} - \frac{1}{2 \cos^{8}{\left(x \right)}} + \frac{1}{10 \cos^{10}{\left(x \right)}}$$

Simplify:

$$\int{\frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x} = \frac{\frac{\cos^{8}{\left(x \right)}}{2} - \cos^{6}{\left(x \right)} + \cos^{4}{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{10}}{\cos^{10}{\left(x \right)}}$$

Add the constant of integration:

$$\int{\frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}} d x} = \frac{\frac{\cos^{8}{\left(x \right)}}{2} - \cos^{6}{\left(x \right)} + \cos^{4}{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{10}}{\cos^{10}{\left(x \right)}}+C$$

$$$\int \frac{\sin^{9}{\left(x \right)}}{\cos^{11}{\left(x \right)}}\, dx = \frac{\frac{\cos^{8}{\left(x \right)}}{2} - \cos^{6}{\left(x \right)} + \cos^{4}{\left(x \right)} - \frac{\cos^{2}{\left(x \right)}}{2} + \frac{1}{10}}{\cos^{10}{\left(x \right)}} + C$$$A