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NDA 2011
Exam
NDA
Year
2011
Subject
Mathematics
Chapter
Indefinite Integration

NDA 2011 · Mathematics · Indefinite Integration

What is ∫ ^ -1 ( x) d x equal to? where k is a constant of...

What is ∫sin⁡−1(cos⁡x)dx\int \sin ^{-1}(\cos x) d x equal to? where kk is a constant of integration.

Options

  1. Axπ2−x22+k\frac{x \pi}{2}-\frac{x^2}{2}+k
  2. Bπ2+x22+k\frac{\pi}{2}+\frac{x^2}{2}+k
  3. C−xπ2−x22+kx-\frac{x \pi}{2}-\frac{x^2}{2}+k x
  4. Dπ2−x22+k\frac{\pi}{2}-\frac{x^2}{2}+k

Correct answer

A. xπ2−x22+k\frac{x \pi}{2}-\frac{x^2}{2}+k

Step-by-step solution

∫sin⁡−1(cos⁡x)dx=∫sin⁡−1{sin⁡(π2)−x}dx\begin{aligned} & \int \sin ^{-1}(\cos x) d x \\ & \quad=\int \sin ^{-1}\left\{\sin \left(\frac{\pi}{2}\right)-x\right\} d x\end{aligned}=∫(π2−x)dx[∴sin⁡−1(sin⁡x)=x]=π2x−x22+c.\begin{aligned} & =\int\left(\frac{\pi}{2}-x\right) d x\left[\therefore \sin ^{-1}(\sin x)=x\right] \\ & =\frac{\pi}{2} x-\frac{x^2}{2}+c.\end{aligned}$
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