Question

How do I evaluate the following integral by parts? `intx^9cos(x^5)dx`

`intx^9cos(x^5)dx`

Answer 1

`I = (x^5sin (x^5) + cos (x^5))/5 + C`

Explanation:

We seek:

`I = int \ x^9 \ cos(x^5) \ dx`

We should first reduce the term `x^5` in the integral by performing a substitution:

Let `\ t =x^5 => (dt)/dx =5x^4`

So we can manipulate the integral, and perform this substitution to get

`I = int \ 1/5 \ x^5 \ cos(x^5) \ (5x^4) \ dx`
`\ \ = 1/5 \ int \ t \ cos t \ du`

And now we can apply Integration By Parts:

Let `{ (u,=t, => (du)/dt,=1), ((dv)/dt,=cost, => v,=sint ) :}`

Then plugging into the IBP formula:

`int \ u \ dv = uv - int v \ du`

We have:

`int \ t \ cost \ dt = (t)(sint) -int \ (sint)(1) \ dt`
`" " = tsin t + cos t \ \ (+c)`

Allowing us to write:

`I = 1/5 {tsin t + cos t} + C`

Then restoring the substitution, we get:

`I = (x^5sin (x^5) + cos (x^5))/5 + C`