AP EAMCET · Maths · Indefinite Integration
If \(\int \frac{d x}{(x \tan x+1)^2}=f(x)+c\), then \(\lim _{x \rightarrow \frac{\pi}{2}} f(x)=\)
- A \(\frac{\pi}{2}\)
- B \(\frac{2}{\pi}\)
- C \(\frac{1}{\pi}\)
- D \(\infty\)
Answer & Solution
Correct Answer
(B) \(\frac{2}{\pi}\)
Step-by-step Solution
Detailed explanation
Let \(f(x) = \frac{\sin x}{x \sin x + \cos x}\). Then \(f'(x) = \frac{\cos x (x \sin x + \cos x) - \sin x (\sin x + x \cos x - \sin x)}{(x \sin x + \cos x)^2} = \frac{\cos^2 x}{(x \sin x + \cos x)^2} = \frac{1}{(x \tan x + 1)^2}\). So,…
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