Let $f(x)=\int \frac{\left(2-x^{2}\right) \cdot e^{x}}{(\sqrt{1+x})(1-x)^{3 / 2}} d x$. If $f(0)=0$, then $\mathrm{f}\left(\frac{1}{2}\right)$ is equal to :

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Indefinite Integrals

Let

f (x) = \int \frac{( 2 - x ^{2} ) \cdot e ^{x}}{( 1 + x ) ( 1 - x ) ^{3/2}} d x

. If

f (0) = 0

, then

f (\frac{1}{2})

is equal to :

Answer: (A)

Rewrite the integrand:

\frac{2 - x ^{2}}{1 + x ( 1 - x ) ^{3/2}} = \frac{( 1 - x ^{2} ) + 1}{1 + x ( 1 - x ) ^{3/2}} = \frac{1 + x}{1 - x} + \frac{1}{1 + x ( 1 - x ) ^{3/2}}

.

Recognize this as

e^{x} (\frac{1 + x}{1 - x})^{'}

, so the integral is

e^{x} \frac{1 + x}{1 - x} + C

.

Using

f (0) = 0

e^{0} \cdot 1 + C = 0

, so

C = - 1

.

Thus

f (x) = e^{x} \frac{1 + x}{1 - x} - 1

.

At

x = \frac{1}{2}

f (\frac{1}{2}) = e \cdot 3 - 1 = 3 e - 1

JEE Main · 2026 · 23 Jan 2026 · Shift 1 · Single correct option

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