Evaluate : _ -8 ^8 x^3 9-x^2 d x - Vidyadip

Question

Evaluate : $\int_{-8}^8 \frac{x^3}{9-x^2} d x$

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Answer

Let $I =\int_{-8}^8 \frac{x^3}{9-x^2} d x$
Now $(x)$$=\frac{x_{k c}^3}{9=x^2}$
$\therefore f(-x)=\frac{(-x)^3}{9-(-x)^2}$
$=\frac{-x^3}{9-x^2}=-f(x)$
$\therefore f(x)$ is an odd function.
$\therefore \int_{-8}^8 \frac{x^3}{9-x^2} d x=0$
$ [\because \int_{-a}^a f(x) d x=0] $ if $f$ is
odd function, theorem.

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