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Question

Mathematics Question on Definite Integral

The value of ∫−1212cos−1xdx\int^{\frac{1}{2}}_{-\frac{1}{2}}cos^{-1}xdx is

A

π\pi

B

π2\frac {\pi}{2}

C

11

D

π22\frac {\pi^2}{2}

Answer

π2\frac {\pi}{2}

Explanation

Solution

∫−1212cos⁡−1xdx\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \cos ^{-1} x d x
=∣xcos⁡−1x∣−1212+∫−1212x1−x2dx=\left|x \cos ^{-1} x\right|_{-\frac{1}{2}}^{\frac{1}{2}}+\int\limits_{-\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\sqrt{1-x^{2}}} d x \quad [Using integration by parts]
=12cos⁡−1(12)+12cos⁡−1(−12)−[(12)⋅21−x2]−1212=\frac{1}{2} \cos ^{-1}\left(\frac{1}{2}\right)+\frac{1}{2} \cos ^{-1}\left(-\frac{1}{2}\right)-\left[\left(\frac{1}{2}\right) \cdot 2 \sqrt{1-x^{2}}\right]_{-\frac{1}{2}}^{\frac{1}{2}}
=12⋅π3+12⋅2π3−0=\frac{1}{2} \cdot \frac{\pi}{3}+\frac{1}{2} \cdot \frac{2 \pi}{3}-0
=π6+π3=π2=\frac{\pi}{6}+\frac{\pi}{3}=\frac{\pi}{2}