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

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

(C) $\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Definite Integrals

The integral

\int_{0}^{\frac{π}{2}} \frac{1}{3 + 2 s i n x + c o s x} d x

is equal to:

Answer: (B)

Using the substitution

t = tan \frac{x}{2}

, we have

sin x = \frac{2 t}{1 + t ^{2}}

cos x = \frac{1 - t ^{2}}{1 + t ^{2}}

, and

d x = \frac{2 d t}{1 + t ^{2}}

.

The integral becomes:

I = \int_{0}^{1} \frac{d t}{( t + 1 ) ^{2} + 1}

This evaluates to:

I = [tan^{- 1} (t + 1)]_{0}^{1} = tan^{- 1} (2) - tan^{- 1} (1) = tan^{- 1} (2) - \frac{π}{4}

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Single correct option

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