Question

${\int }_{-9}^9{x}^{99}dx$ = ___

Answer 1

The given problem is one of those which have their appearance quite scary but actually are very easy to solve. We can see that the limits given are symmetric to y- axis i.e. (-a to a). We know how to solve these problems, we’ll check whether the function is even or odd and use the following property.
$\left(i\right){\int }_{-a}^{a}f\left(x\right)dx=2{\int }_0^{a}f\left(x\right)dx$,if f is even,i.e.,$iff(-x)=f\left(x\right)$
$\left(ii\right){\int }_{-a}^{a}f\left(x\right)dx=0,$ if f is odd,i.e.,$iff(-x)=-f\left(x\right)$.
Let f(x) be $=x^{99}$
Let’s check whether the function is even or odd.
$f(-x)=(-x{)}^{99}$
$f(-x)=-x^{99}=-f\left(x\right)$
So, the given function is odd.
And hence, ${\int }_{-9}^9{x}^{99}dx=0$