Math Help - Integration using given substitution difficulty.

1. Integration using given substitution difficulty.

Hello again,

Just when I think I'm getting it I realize I'm really not.

I have no idea how to approach this question, I have a bunch like this to do and would really appreciate some help.

The question is, using the substitution:

$u^2 = 1 + \tan(x)$

Evaluate the integral:

$\int\sec^2(x)\tan(x)\sqrt{1 + \tan(x)} dx$

I know that:

$\tan(x) = u^2 - 1$ that $\sec^2(x) = \tan^2(x) + 1$ and that $u = \sqrt{1 + \tan(x)}$ but have no idea how to proceed. I have looked in all my text books but can't't find an example similar enough to help me figure out what to do.

Thank you.

2. Re: Integration using given substitution difficulty.

Use implicit differentiation to find du/dx
From
$u^2 = 1 + \tan(x)$
$2u$ $du/dx = \sec^2(x)$

Replace
$\sec^2(x)$ $dx$ with $2u$ $du$
And replace $\tan(x)$ with $u^2 -1$