Question

Prove the following question.${\int }_{-1}^1{x}^{17}co{s}^{4}xdx=0.$

Answer 1

Let ${\int }_{-1}^1{x}^{17}co{s}^{4}xdx=0.\Rightarrow f(-x)=(-x{)}^{17}co{s}^4(-x)=-x^{17}co{s}^{4}x=-f(x)$

Therefore, f(x) is an odd function.

We know that if f(x) is an odd function, then ${\int }_{-a}^{a}f\left(x\right)dx=0$

$\therefore {\int }_{-1}^1{x}^{17}co{s}^{4}xdx=0.$Hence proved.

Note If f(x) is odd, then ${\int }_{-a}^{a}f\left(x\right)dx=0$

If f(x) is even, then ${\int }_{-a}^{a}f\left(x\right)dx=2{\int }_a^{a}f\left(x\right)dx$

And if f(x) is neither odd nor even, then we do not apply any rule.