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

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

1 Answer
Steve M
May 22, 2017

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

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