Question

How do you integrate `(x^(n+1))/(n+1)?`

I know how to integrate `x^n`, but I don't understand how to integrate the result.

Answer 1

`int \ x^(n+1)/(n+1) \ dx = x^(n+2)/((n+1)(n+2)) + C`

Explanation:

We use the standard result:

`int \ x^n \ dx = x^(n+1)/(n+1) + C`

Or to avoid confusion, let us write it as:

`int \ x^m \ dx = x^(m+1)/(m+1) + C`

So with `m=n+1` we have:

`int \ x^(n+1)/(n+1) \ dx = 1/(n+1) int \ x^(n+1) \ dx`
`" " = 1/(n+1) int \ x^m \ dx`
`" " = 1/(n+1) x^(m+1)/(m+1) + C`
`" " = 1/(n+1) x^(n+1+1)/(n+1+1) + C`
`" " = 1/(n+1) x^(n+2)/(n+2) + C`
`" " = x^(n+2)/((n+1)(n+2)) + C`