The integral _ 0 ^ 2 1 3+2 x+ x d x is equal to. - Vidyadip

MCQ

The integral $\int_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} d x$ is equal to.

A
$\tan ^{-1}(2)$
✓
$\tan ^{-1}(2)-\frac{\pi}{4}$
C
$\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$
D
$\frac{1}{2}$

✓

Answer

Correct option: B.

$\tan ^{-1}(2)-\frac{\pi}{4}$

b
$I=\int_{0}^{\frac{\pi}{2}} \frac{d x}{3+2 \sin x+\cos x}=\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \frac{x}{2} \cdot d x}{2 \tan ^{2} \frac{x}{2}+4 \tan \frac{x}{2}+4}$

Put $\tan \frac{x}{2}=t$, so

$I=\int_{0}^{1} \frac{d t}{(t+1)^{2}+1}=\left.\tan ^{-1}(x+1)\right|_{0} ^{1}=\tan ^{-1} 2-\frac{\pi}{4}$

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