JEE Main 2017 · Mathematics · Indefinite and Definite Integrals · medium

The value of ∫-1^1(x^2/(1+e^x dx is

Single correct option

The value of

\displaystyle\int_{-1}^{1}\dfrac{x^2}{1+e^{x}}\,dx

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Answer: (C) $\dfrac13$

Let

I=\int_{-1}^1\frac{x^2}{1+e^x}dx

. Replacing

x

-x

gives

I=\int_{-1}^1\frac{x^2e^x}{1+e^x}dx

. Adding the two forms,

2I=\int_{-1}^1x^2\,\frac{1+e^x}{1+e^x}dx=\int_{-1}^1x^2dx=\dfrac23

. Therefore

I=\dfrac13

. The integrand is neither even nor odd, so the symmetry has to be used in this paired form.

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