Unraveling a Challenging Integral: Solving for x-arctan(x)+c

[sckeen1988]

1. The Problem comes down to what I think is a fairly simple integral, I just can't figure it out.

int(x^2/(1+x^2)

I know this is x-arctan(x)+c, but cannot figure out how.

I tried subbing in u for x^2, this didn't really help me with a direction to take though.

 

[genericusrnme]

if you ever see a 1+x^2 you should imediately think to yourself TRIG SUBSTITUTION!

What relation do you know between sin^2 and cos^2?

I'd start there

 

[Ray Vickson]

1. The Problem comes down to what I think is a fairly simple integral, I just can't figure it out.

int(x^2/(1+x^2)

I know this is x-arctan(x)+c, but cannot figure out how.

I tried subbing in u for x^2, this didn't really help me with a direction to take though.

First, write f(x) = x^2/(1+x^2) as f(x) = (x^2 + 1 - 1)/(1+x^2) = 1 - 1/(1+x^2), so
int(f(x)) = x - int(1/(1 + x^2)) The expression 1 + x^2 cries out for the substitution x = tan(t).

RGV

 

[sckeen1988]

First, write f(x) = x^2/(1+x^2) as f(x) = (x^2 + 1 - 1)/(1+x^2) = 1 - 1/(1+x^2), so
int(f(x)) = x - int(1/(1 + x^2)) The expression 1 + x^2 cries out for the substitution x = tan(t).

RGV

ANGER! I knew I was forgetting something really easy, like adding and subtracting one. Thanks for the help