Please help me with a trig derivative

**phack** · January 30th 2007, 01:17 PM

I need to find the integral of this:

Any help would be appreciated!

**ThePerfectHacker** · January 30th 2007, 01:38 PM

Originally Posted by phack

I need to find the integral of this:

Any help would be appreciated!

You need to find,
$\int \frac{\sec^4 \sqrt{x}}{\sqrt{x}} dx$
Let, $u=\sqrt{x}$
Thus, after the substitution,
$2\int \sec^4 u du$
Now write,
$2\int \sec^2 u \sec^2 u du=2\int (\tan^2 +1)\sec^2 u du$
Let, $t=\tan u$ then $t'=\sec^2 u$ .
Thus,
$2\int (t^2+1) dt=\frac{2}{3}t^3+2t+C$
Substitute,
$\frac{2}{3} \tan^3 u+2\tan u+C$
Substitute,
$\frac{2}{3} \tan^3 \sqrt{x}+2\tan \sqrt{x}+C$

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