JEE Main202623 January 2026Morning ShiftMathematicsIndefinite IntegrationActual

Let f(x)= (2-x² ) e ^ x ( 1+x )(1-x)^ 3 / 2 ~d x . If f(0)=0 , then f ( 1 2 ) is equal to:

Options

A3 e -1
B3 e +1
C2 e +1
D2 e -1

Correct answer

A. 3 e -1

Step-by-step solution

Write 2 - x^2 = (1-x)(1+x) + 1 , so the integrand splits as e^x ( 1+x 1-x + 1 1+x (1-x)^ 3/2 ) . Let g(x) = 1+x 1-x . Then g'(x) = 1 1+x (1-x)^ 3/2 . Using e^x(g + g')dx = e^x g + C : f(x) = e^x 1+x 1-x + C . f(0) = 1 + C = 0 C = -1 . f(1/2) = e^ 1/2 3/2 1/2 - 1 = e 3 - 1 = 3e - 1 .

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