Question

How do you evaluate the definite integral `int (x^45)(cos(x^46)) dx` from `[0,(pi)^(1/46)]`?

Answer 1

`0`

Explanation:

We have

`int_0^(pi^(1//46))x^45cos(x^46)dx`

Substituting, let `u=x^46` and `du=46x^45dx`.

Multiply the interior of the integral by `46` and the exterior by `1//46`.

`=1/46int_0^(pi^(1//46))cos(x^46)*46x^45dx`

Now substitute in for `u`. Recall that using `u` substitution will cause the bounds of the definite integral to change. We can find the new bound by plugging the current bounds into `u=x^46`.

`"bound of"` `0->" "0^46=0" "larr"new bound"`

`"bound of"` `pi^(1//46)->" "(pi^(1//46))^46=pi" "larr"new bound"`

This gives us the integral of

`=1/46int_0^picos(u)du`

Which then becomes

`=1/46[sin(u)]_0^pi=1/46(sin(pi)-sin(0))=1/46(0-0)=0`