Question

$\int \frac{\text{cosec}^2 x-2017}{\cos ^{2017} x} d x$ is equal to

• A. $ -\frac{\text{cosec} x}{(\cos x)^{2016}}+C$
• B. $\frac{\cot x}{(\cos x)^{2017}}+C$
• C. $-\frac{\cot x}{(\cos x)^{2016}}$
• D. $\frac{\tan x}{(\cos x)^{2017}}+C$

Answer 1

Answer: (A)

$I=\int(\cos \overset{I}{x})^{-2017} \cdot \overset{II}{cosec^2 x} d x-2017 \int(\cos x)^{-2017} d x$
$=(-\cot x) \cdot(\cos x)^{-2017}-\int(-2017) \cdot(\cos x)^{-2018} \cdot(-\sin x)(-\cot x) d x-2017 \int(\cos x)^{-2017} d x$
$I=\frac{-\cot x}{(\cos x)^{2017}}+2017 \int(\cos x)^{-2017} d x-\int(2017)(\cos x)^{-2017} d x$
$I=-\frac{\operatorname{cosec} x}{(\cos x)^{2016}}+C$