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GUJCET · Maths · Integrals
\(\int e^x\left(\frac{1+\sin x}{1+\cos x}\right) d x=\) ____________ \(+c\).
- A \(e^x \tan x\)
- B \(e^x \tan \frac{x}{2}\)
- C \(e^x \cot \frac{x}{2}\)
- D \(e^x \cot x\)
Answer & Solution
Correct Answer
(B) \(e^x \tan \frac{x}{2}\)
Step-by-step Solution
Detailed explanation
\(\int e^x\left(\frac{1+\sin x}{1+\cos x}\right) d x = \int e^x\left(\frac{1}{1+\cos x} + \frac{\sin x}{1+\cos x}\right) d x\) \(= \int e^x\left(\frac{1}{2\cos^2\left(\frac{x}{2}\right)} + \frac{2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)}{2\cos^2\left(\frac{x}{2}\right)}\right) d x\)
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