AP EAMCET · Maths · Indefinite Integration
\(\int \frac{x+\sin x}{1+\cos x} d x=\)
- A \(x \tan \frac{x}{2}+C\)
- B \(x \sin \frac{x}{2}+\cos \frac{x}{2}+C\)
- C \(x \tan \frac{x}{2}+\sec \frac{x}{2}+C\)
- D \(x \sec \frac{x}{2}+\tan \frac{x}{2}+C\)
Answer & Solution
Correct Answer
(A) \(x \tan \frac{x}{2}+C\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { Let } I=\int \frac{x+\sin x}{1+\cos x} d x \\ & =\int \frac{x}{1+\cos x} d x+\int \frac{\sin x}{1+\cos x} d x \\ & =\int \frac{x}{2 \cos ^2 \frac{x}{2}} d x+\int \frac{\sin x}{2 \cos ^2 \frac{x}{2}} d x \\ & =\frac{1}{2} \int x \sec ^2 \frac{x}{2} d…
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