MHT CET · Maths · Indefinite Integration
The value of \(\int(1-\cos x) \cdot \operatorname{cosec}^2 x \mathrm{~d} x\) is
- A \(\frac{1}{2} \tan \frac{x}{2}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- B \(\tan \frac{x}{2}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- C \(2 \cot \frac{x}{2}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
- D \(\cot \frac{x}{2}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Answer & Solution
Correct Answer
(B) \(\tan \frac{x}{2}+\mathrm{c}\), where \(\mathrm{c}\) is a constant of integration.
Step-by-step Solution
Detailed explanation
Let \(\mathrm{I}=\int(1-\cos x) \cdot \operatorname{cosec}^2 \mathrm{~d} x\)
\(=\int \frac{2 \sin ^2 \frac{x}{2}}{[\sin x]^2} \mathrm{~d} x \quad \ldots\left[\because 1-\cos \theta=2 \sin ^2 \frac{\theta}{2}\right]\)
\(\begin{aligned} &=\int \frac{2 \sin ^2 \frac{x}{2}}{\left[2 \sin \frac{x}{2} \cos \frac{x}{2}\right]^2} \mathrm{~d} x \\ & \ldots\left[\because \sin \theta=2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\right]\end{aligned}\)
\(\begin{aligned}
& =\frac{1}{2} \int \sec ^2 \frac{x}{2} \mathrm{~d} x \\
& =\tan \frac{x}{2}+\mathrm{c}
\end{aligned}\)
\(=\int \frac{2 \sin ^2 \frac{x}{2}}{[\sin x]^2} \mathrm{~d} x \quad \ldots\left[\because 1-\cos \theta=2 \sin ^2 \frac{\theta}{2}\right]\)
\(\begin{aligned} &=\int \frac{2 \sin ^2 \frac{x}{2}}{\left[2 \sin \frac{x}{2} \cos \frac{x}{2}\right]^2} \mathrm{~d} x \\ & \ldots\left[\because \sin \theta=2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}\right]\end{aligned}\)
\(\begin{aligned}
& =\frac{1}{2} \int \sec ^2 \frac{x}{2} \mathrm{~d} x \\
& =\tan \frac{x}{2}+\mathrm{c}
\end{aligned}\)
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