The value of $\int_{-1}^{1} \frac{(1+\sqrt{|x|-x}) e^{x}+(\sqrt{|x|-x}) e^{-x}}{e^{x}+e^{-x}} d x$ is equal to

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Definite Integrals

The value of

\int_{- 1}^{1} \frac{( 1 + ∣ x ∣ - x ) e ^{x} + ( ∣ x ∣ - x ) e ^{- x}}{e ^{x} + e ^{- x}} d x

is equal to

Answer: (D)

Replacing

x

- x

turns

∣ x ∣ - x

into

∣ x ∣ + x

and swaps

e^{x}

with

e^{- x}

; adding this to the original integral, the factor

e^{x} + e^{- x}

cancels and

2 I = \int_{- 1}^{1} (1 + ∣ x ∣ - x + ∣ x ∣ + x) d x

. Here

∣ x ∣ + x = 2 x

[0, 1]

and

∣ x ∣ - x = - 2 x

[- 1, 0]

, each integrating to

\frac{2 2}{3}

, while

\int_{- 1}^{1} 1 d x = 2

. Thus

2 I = 2 + \frac{4 2}{3}

, so

I = 1 + \frac{2 2}{3}

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Single correct option

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