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:

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

Evaluate the integral:

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

I know that:

$\displaystyle \tan(x) = u^2 - 1$ that $\displaystyle \sec^2(x) = \tan^2(x) + 1$ and that $\displaystyle u = \sqrt{1 + \tan(x)}$ 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.

Re: Integration using given substitution difficulty.

Use implicit differentiation to find du/dx

From

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

$\displaystyle 2u$ $\displaystyle du/dx = \sec^2(x)$

Replace

$\displaystyle \sec^2(x)$ $\displaystyle dx$ with $\displaystyle 2u$ $\displaystyle du$

And replace $\displaystyle \tan(x)$ with $\displaystyle u^2 -1$