Question

Evaluate the following :​

Answer 1

Question :- Evaluate the following :

$\rm \: \displaystyle\int_{-8}^{8}\rm \frac{ {x}^{3} }{9 - {x}^{2} } \: dx$

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm \: \displaystyle\int_{-8}^{8}\rm \frac{ {x}^{3} }{9 - {x}^{2} } \: dx$

We know,

$\boxed{ \rm{ \:\begin{gathered}\begin{gathered}\displaystyle\int_{-a}^{a}\rm f(x)dx = \begin{cases} &\sf{0 \: \: if \: f( - x) = - f(x)} \\ \\ &\sf{2\displaystyle\int_{0}^{a}\rm f(x)dx \: \: if \: f( - x) = f(x)} \end{cases}\end{gathered}\end{gathered}}}$

So, Consider

$\rm \: f(x) = \dfrac{ {x}^{3} }{9 - {x}^{2} }$

So,

$\rm \: f( - x) = \dfrac{ {( - x)}^{3} }{9 - {( - x)}^{2} }$

$\rm \: f( - x) = \dfrac{ - {x}^{3} }{9 - {x}^{2} }$

$\rm \: f( - x) \: = \: - \: \dfrac{{x}^{3} }{9 - {x}^{2} }$

$\rm\implies \:f( - x) \: = \: - \: f(x)$

Hence, using above property of definite integrals, we have

$\rm\implies \:\boxed{ \rm{ \:\rm \: \displaystyle\int_{-8}^{8}\rm \frac{ {x}^{3} }{9 - {x}^{2} } \: dx = 0 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{ \rm{ \:\displaystyle\int_{a}^{b}\rm f(x) \: dx \: = \: \displaystyle\int_{a}^{b}\rm f(y) \: dy \: \: }}$

$\boxed{ \rm{ \:\displaystyle\int_{a}^{b}\rm f(x) \: dx \: = \: - \: \displaystyle\int_{b}^{a}\rm f(x) \: dx \: \: }}$

$\boxed{ \rm{ \:\displaystyle\int_{0}^{a}\rm f(x) \: dx \: = \: \displaystyle\int_{0}^{a}\rm f(a - x) \: dx \: \: }}$

$\boxed{ \rm{ \:\displaystyle\int_{a}^{b}\rm f(x) \: dx \: = \: \displaystyle\int_{a}^{b}\rm f(a + b- x) \: dx \: \: }}$

$\boxed{ \rm{ \:\displaystyle\int_{0}^{2a}\rm f(x) \: dx \: = 2 \: \displaystyle\int_{0}^{a}\rm f(x) \: dx \: if \: f(2a - x) = f(x) \: }}$

$\boxed{ \rm{ \:\displaystyle\int_{0}^{2a}\rm f(x) \: dx \: = 0 \: if \: f(2a - x) = - f(x) \: }}$

Answer 2

refer the given attachment