AP EAMCET · Maths · Indefinite Integration
\(\int e^{x / 2}\left(\frac{2+\sin x}{1+\cos x}\right) d x=\)
- A \(2 e^{x / 2} \operatorname{cosec}\left(\frac{x}{2}\right)+c\)
- B \(2 e^{x / 2} \tan \left(\frac{x}{2}\right)+c\)
- C \(2 e^{x / 2} \cos \left(\frac{x}{2}\right)+c\)
- D \(2 e^{x / 2} \sin \left(\frac{x}{2}\right)+c\)
Answer & Solution
Correct Answer
(B) \(2 e^{x / 2} \tan \left(\frac{x}{2}\right)+c\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \int e^{\frac{x}{2}}\left(\frac{2+\sin x}{1+\cos x}\right) d x \\ & \int e^{\frac{x}{2}}\left(\frac{2+\frac{2 \tan x / 2}{1+\tan ^2 x / 2}}{1+\frac{1-\tan ^2 x / 2}{1+\tan ^2 x / 2}}\right) d x \\ & \int e^{\frac{x}{2}}\left(\frac{2+2 \tan ^2 x / 2+2 \tan x /…
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