# $sec^{2}(x)tan(x)dx$ Let U = sec(x) or tan(x) ?

The integral of $sec^{2}(x)tan(x)dx$ is what I'm asking about.

## Homework Statement

$sec^{2}(x)tan(x)dx$

I can let u = tanx
then du = $sec^{2}(x)$

$\frac{1}{2}\int udu$

$\frac{1}{2}tan^{2}(x)$

OR

I can let u = secx
then du = secxtanx dx

$\int udu$

$\frac{1}{2}u^{2}$

$\frac{1}{2}sec^{2}(x)$

Why do both work? Which one is correct? Or are both correct?

Dick
Homework Helper
The integral of $sec^{2}(x)tan(x)dx$ is what I'm asking about.

## Homework Statement

$sec^{2}(x)tan(x)dx$

I can let u = tanx
then du = $sec^{2}(x)$

$\frac{1}{2}\int udu$

$\frac{1}{2}tan^{2}(x)$

OR

I can let u = secx
then du = secxtanx dx

$\int udu$

$\frac{1}{2}u^{2}$

$\frac{1}{2}sec^{2}(x)$

Why do both work? Which one is correct? Or are both correct?
They both work. They differ by a constant. sec(x)^2-tan(x)^2=1. You should put a '+C' in when you write the solution to an indefinite integral. That's where the difference is.

thanks, appreciate the feedback.