WBJEE · Maths · Definite Integration
The value of integral \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{(sinx-xcosx)}{x(x+sinx)}dx\) is
- A \(\log _{e}\left\{\frac{2(\pi+3)}{(2 \pi+3 \sqrt{3})}\right\}\)
- B \(\log _{e}\left\{\frac{\pi+3}{2(2 \pi+3 \sqrt{3})}\right\}\)
- C \(\log _{e}\left\{\frac{2 \pi+3 \sqrt{3}}{2(\pi+3)}\right\}\)
- D \(\log _{e}\left\{\frac{2(2 \pi+3 \sqrt{3})}{\pi+3}\right\}\)
Answer & Solution
Correct Answer
(A) \(\log _{e}\left\{\frac{2(\pi+3)}{(2 \pi+3 \sqrt{3})}\right\}\)
Step-by-step Solution
Detailed explanation
Let \(l=\int_{\pi / 6}^{\pi / 3} \frac{(\sin x-x \cos x)}{x(x+\sin x)} d x\) \(\Rightarrow \quad I=\int_{\pi / 6}^{\pi / 3} \frac{(x+\sin x)-x(1+\cos x)}{x(x+\sin x)} d x\) \(\Rightarrow \quad I=\int_{\pi / 6}^{\pi / 3}\left(\frac{1}{x}-\frac{1+\cos x}{x+\sin x}\right) d x\)…
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