Evaluate: ^ 2 _ -2 x ^ 2 1 + 5^ x dx. - Vidyadip

Question

Evaluate: $\int\limits^{2}_{-2} \frac{{x}^{2}}{1 + 5^{x}} dx. $

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Answer

Using property$:\int\limits^{\text{b}}_{a}\text{f(x) dx} = \int\limits^{\text{b}}_{\text{a}} \text{f(a + b }{-}{\text{x}) \text{dx}}$
$\text{I} = \int\limits^{2}_{-2}\bigg(\frac{\text{x}^{2}}{\text{1 + 5}^{\text{x}}}\bigg) \text{dx} = \int\limits^{2}_{-2}\bigg(\frac{\text{x}^{2}}{\text{1 + 5}^{\text{-x}}}\bigg) \text{dx}$
$\text{2I} = \int\limits^{2}_{-2}\text{x}^{2}\text{dx}$
$\text{2I} = \frac{16}{3} \text{or I} = \frac{8}{3}$

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