Integrating Sin(x)/Cos^2(x) using u-substitution

[PauloE]

Homework Statement

$∫sec(x)tan(x)+x/(x^2+1) dx$

The Attempt at a Solution

I replaced sec and tan by $1/cos(x)$ and $sin/cos(x)$ then end up with $sin(x)/cos^2(x)$
then I replace $cos^2 x$ by $1-sin^2 x$ then I don't know where to go. the second part of the equation works with u substitution.
I just can't see where the identities of the first part are leading me.

any hint? thanks in advance!

 

[Curious3141]

Homework Statement

$∫sec(x)tan(x)+x/(x^2+1) dx$

The Attempt at a Solution

I replaced sec and tan by $1/cos(x)$ and $sin/cos(x)$ then end up with $sin(x)/cos^2(x)$
then I replace $cos^2 x$ by $1-sin^2 x$ then I don't know where to go.

No, don't do that. Try the sub $u = \cos x$ and watch that sucker fold.

 

[PauloE]

you know i just used tan(x) in the first term and u substitution in the second and it worked too!

Thanks a lot!
Paulo

 

[vela]

You should ideally recognize the derivative of sec x.

 

[utkarshakash]

For the first integral you don't have to substitute anything. It is the derivative of sec(x).