Q102 / 180 · COMEDK 2017 · Mathematics
Evaluate $\int_{0}^{\pi / 2} \frac{\cos x}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{3}} d x$.
- Correct answerA$2-\sqrt{2}$
- B$2+\sqrt{2}$
- C$3+\sqrt{3}$
- D$3-\sqrt{3}$
Solution
$\mathrm{We}$ have,
$=\int_{0}^{\pi / 2} \frac{\cos \frac{x}{2}-\sin \frac{x}{2}}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^{2}} d x$
Put $\cos \frac{x}{2}+\sin \frac{x}{2}=t$
$\Rightarrow \frac{1}{2}\left(-\sin \frac{x}{2}+\cos \frac{x}{2}\right) d x=d t$
Also, when $x=0 \Rightarrow t=1$
When $x=\frac{\pi}{2} \Rightarrow t=\sqrt{2}$
$\begin{aligned}
\therefore \quad I &=\int_{1}^{\sqrt{2}} \frac{2 d t}{t^{2}} \\
&=2\left[-\frac{1}{t}\right]_{1}^{\sqrt{2}}=2\left[-\frac{1}{\sqrt{2}}+1\right]=(2-\sqrt{2})
\end{aligned}$
Hence, the correct answer is option: 'A'.