Practicing Success
If $\int \frac{1-7 \cos ^2 x}{\sin ^7 x \cos ^2 x} d x=\frac{f(x)}{\sin ^7 x}+C$ is equal to |
$\sin x$ $\cos x$ $\tan x$ $\cot x$ |
$\tan x$ |
Let $I=\int \frac{1-7 \cos ^2 x}{\sin ^7 x \cos ^2 x} d x$ $\Rightarrow I=\int \frac{\sec ^2 x}{\sin ^7 x} d x-7 \int \frac{1}{\sin ^7 x} d x$ $\Rightarrow I=\int \frac{\tan x}{\sin ^7 x}-\int(-7)(\sin x)^{-8} \cos x \tan x d x-7 \int \frac{1}{\sin ^7 x} d x$ $\Rightarrow I=\int \frac{\tan x}{\sin ^7 x}+7 \int \frac{1}{\sin ^7 x} d x-7 \int \frac{1}{\sin ^7 x} d x+C$ $\Rightarrow I=\frac{\tan x}{\sin ^7 x}+C$ Hence, $f(x)=\tan x$. |