Question

The value of the integration $\int\limits^{ \pi/4}_{- \pi/4}$$\left(\lambda \left|sin\,x\right|+\frac{\mu\, sin \,x}{1+cos\,x}+\gamma\right)dx$

• A. is independent of $\lambda$ only
• B. is independent of $\mu$ only
• C. is independent of $\gamma$ only
• D. depends on $\lambda$ $\mu$ and $\gamma$

Answer 1

Answer: (B)

$I = 2\lambda \int\limits^{\frac{\pi}{4}}_{0} sin\,x\,dx + \mu \int\limits^{\frac{ \pi}{4}}_{-\frac{\pi}{4}} tan \frac{x}{2}dx+\gamma \int\limits^{\frac{ \pi}{4}}_{-\frac{\pi}{4}} dx$
$= 2\lambda\left(1-\frac{1}{\sqrt{2}}\right)-0+\gamma\left(\frac{\pi}{2}\right)\quad\quad\quad$($\because \, tan \frac{x}{2}$ is an odd function)