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CBSE Class 12 · 2016 · Definite IntegralsLong Answer

Evaluate:

\int_{- 2}^{2} \frac{x ^{2}}{1 + 5 ^{x}} d x

Answer: $\frac{8}{3}$

Solution

Let

I = \int_{- 2}^{2} \frac{x ^{2}}{1 + 5 ^{x}} d x

. Apply

\int_{- a}^{a} f (x) d x = \int_{- a}^{a} f (- x) d x

:

I = \int_{- 2}^{2} \frac{x ^{2}}{1 + 5 ^{- x}} d x = \int_{- 2}^{2} \frac{5 ^{x} x ^{2}}{1 + 5 ^{x}} d x

Add the two forms; the denominators combine to

1

:

2 I = \int_{- 2}^{2} x^{2} d x = 2 \int_{0}^{2} x^{2} d x = 2 \cdot \frac{8}{3} = \frac{16}{3}

Hence

I = \frac{8}{3}

.

CBSE Class 12 Board · 2016 · Subjective

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