Search any question & find its solution
Question:
Answered & Verified by Expert
$\int(1-\cos x) \operatorname{cosec}^2 x d x$ is equal to
Options:
Solution:
1385 Upvotes
Verified Answer
The correct answer is:
$\tan \left(\frac{x}{2}\right)+c$
Let $I=\int(1-\cos x) \operatorname{cosec}^2 x d x$
$I=\int \operatorname{cosec}^2 x d x-\int \cos x \cdot \operatorname{cosec}^2 x d x$
$\begin{aligned} & =-\cot x-\int \cos x \cdot \operatorname{cosec} x d x \\ & =-\cot x+\operatorname{cosec} x+C\end{aligned}$
$=\frac{1-\cos x}{\sin x}+C$ $\left[\because 1-\cos x=2 \sin ^2 \frac{x}{2}\right]$
$=\frac{2 \sin ^2 x / 2}{2 \sin x / 2 \cdot \cos x / 2}+C$
$I=\tan \left(\frac{x}{2}\right)+C$
$I=\int \operatorname{cosec}^2 x d x-\int \cos x \cdot \operatorname{cosec}^2 x d x$
$\begin{aligned} & =-\cot x-\int \cos x \cdot \operatorname{cosec} x d x \\ & =-\cot x+\operatorname{cosec} x+C\end{aligned}$
$=\frac{1-\cos x}{\sin x}+C$ $\left[\because 1-\cos x=2 \sin ^2 \frac{x}{2}\right]$
$=\frac{2 \sin ^2 x / 2}{2 \sin x / 2 \cdot \cos x / 2}+C$
$I=\tan \left(\frac{x}{2}\right)+C$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.