MHT CET · Maths · Definite Integration
\(\int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} d x=\)
- A \(\frac{\pi}{2}\)
- B \(\frac{\pi}{4}\)
- C \(\frac{\pi}{3}\)
- D \(\frac{\pi}{6}\)
Answer & Solution
Correct Answer
(D) \(\frac{\pi}{6}\)
Step-by-step Solution
Detailed explanation
\(\int_0^1 \frac{1}{\sqrt{3+2 x-x^2}} \mathrm{~d} x=\int_0^1 \frac{\mathrm{d} x}{\sqrt{(2)^2-(x-1)^2}}=\) \(\left[\sin ^{-1}\left(\frac{x-1}{2}\right)\right]_0^1 \)
\( =\sin ^{-1}(0)-\sin ^{-1}\left(\frac{-1}{2}\right)=0-\left(-\frac{\pi}{6}\right)=\frac{\pi}{6}\)
\( =\sin ^{-1}(0)-\sin ^{-1}\left(\frac{-1}{2}\right)=0-\left(-\frac{\pi}{6}\right)=\frac{\pi}{6}\)
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