The value of integration _ - 4 ^ 4 ( | x|+ x 1+ x + ) d x - Vidyadip

MCQ

The value of integration $\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\left(\lambda|\sin x|+\frac{\mu \sin x}{1+\cos x}+\gamma\right) d x$

A
is independent of $\lambda$ only
✓
is independent of $\mu$ only
C
is independent of $\gamma$ only
D
depends on $\lambda, \mu$ and $\gamma$

✓

Answer

Correct option: B.

is independent of $\mu$ only

(B)
Let $I =\int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\left(\lambda|\sin x|+\frac{\mu \sin x}{1+\cos x}+\gamma\right) d x$
$=\lambda \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}|\sin x| d x+\mu \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}}\left(\frac{\sin x}{1+\cos x}\right) d x+\gamma \int_{\frac{-\pi}{4}}^{\frac{\pi}{4}} 1 d x$
Here, $|\sin x|$ and 1 are even functions and $\left(\frac{\sin x}{1+\cos x}\right)$ is an odd function.
$I =2 \lambda \int_0^{\frac{\pi}{4}} \sin x d x+0+2 \gamma \int_0^{\frac{\pi}{4}} 1 d x$
$\begin{array}{l}=2 \lambda[-\cos x]_0^{\frac{\pi}{4}}+2 \gamma[x]_0^{\frac{\pi}{4}} \\ =2 \lambda\left(-\frac{1}{\sqrt{2}}+1\right)+2 \gamma\left(\frac{\pi}{4}-0\right),\end{array}$
which is independent of $\mu$

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