AP EAMCET · Maths · Indefinite Integration
If \(\int \frac{x}{x \tan x+1} \mathrm{~d} x=\log \mathrm{f}(x)+\mathrm{k}\), then \(\mathrm{f}\left(\frac{\pi}{4}\right)=\)
- A \(\frac{\pi}{4 \sqrt{2}}\)
- B \(\pi+\frac{\pi}{2 \sqrt{2}}\)
- C \(\frac{\pi+4}{4 \sqrt{2}}\)
- D \(\frac{\pi-4}{4 \sqrt{2}}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi+4}{4 \sqrt{2}}\)
Step-by-step Solution
Detailed explanation
\(\int \frac{x}{x \tan x+1} \mathrm{~d} x = \int \frac{x \cos x}{x \sin x + \cos x} \mathrm{~d} x\) \(\int \frac{x \cos x}{x \sin x + \cos x} \mathrm{~d} x = \log |x \sin x + \cos x| + \mathrm{k}\) \(\mathrm{f}(x) = x \sin x + \cos x\)…
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