AP EAMCET · Maths · Definite Integration
\(\int_0^{\pi / 2} \log _e(\sin 2 x) d x\)
- A \(\pi \log 2\)
- B \(-\pi \log 2\)
- C \(\frac{\pi}{2} \log 2\)
- D \(-\frac{\pi}{2} \log 2\)
Answer & Solution
Correct Answer
(D) \(-\frac{\pi}{2} \log 2\)
Step-by-step Solution
Detailed explanation
\(I=\int_0^{\pi / 2} \log _e(\sin 2 x) d x\) Let \[ 2 x=t \] For lower limit at \(x=0, t=0\) and upper limit at \(x=\pi / 2, t=\pi\) and \(d x=\frac{1}{2} d t\) So, \(I=\frac{1}{2} \int_0^\pi \log _e \sin (t) d t=\int_0^{\pi / 2} \log _e \sin (t) d t\)…
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