integrals using substution?

Question

evaluate the integral of

sec^2xtanxdx, u=tanx

can someone explain how to do this?

I'd appreciate any help.

Thanks.

Answer 1

You know u=tanx.

Take the derivative:

du = sec^2x dx

Now substitute into the original equation.

The tanx becomes u, and the sec^2x dx becomes du:

u du

Integrate:

(1/2)u^2 + C

Substitute back in u=tanx:

(1/2)tan^2x + C

Answer 2

let u = tanx du=sec^2xdx

so the integrals tanx sec^2xdx becomes udu
= u^2/2 + c
= 1/2 tan^2x + c