- #1

Char. Limit

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## Homework Statement

So, for absolutely no good reason I decided to practice trigonometric substitution. I started with [itex]\int \left(1-x^2\right)^{-1/2} dx[/itex], and that was easy, everything cancelled out nicely. Then I tried [itex]\int \left(1+x^2\right)^{-1/2} dx[/itex], and although the integral was fine, everything did not cancel out nicely afterward. So I need help on one minor issue.

## Homework Equations

[tex] sec(u) = \frac{sec^2(u) + sec(u) tan(u)}{sec(u) + tan(u)}[/tex]

## The Attempt at a Solution

[tex]\int \frac{dx}{\sqrt{1+x^2}}[/tex]

[tex]\int \frac{sec^2(u) du}{\sqrt{1+tan^2(u)}}[/tex]

[tex]\int sec(u) du[/tex]

[tex]\int \frac{sec^2(u) + sec(u) tan(u)}{sec(u) + tan(u)} du[/tex]

[tex]\int \frac{dv}{v}[/tex]

[tex]ln(v)[/tex]

[tex]ln(tan(u) + sec(u))[/tex]

[tex]ln(x + sec(tan^{-1}(x)))[/tex]

Now, is there any way to simplify sec(arctan(x))?