# Trigonometric Substitution Question

• Jan 21st 2010, 08:24 PM
Troopa
Trigonometric Substitution Question
Hello,

I am absolutely stumped on this one.

http://www.mario-kart.net/gamespy/im...Untitled-3.gif

I'm assuming all of my work is correct up until the end, but I don't know how to proceed using Trigonometric Substitution. Please help!
• Jan 21st 2010, 08:37 PM
Jhevon
Quote:

Originally Posted by Troopa
Hello,

I am absolutely stumped on this one.

http://www.mario-kart.net/gamespy/im...Untitled-3.gif

I'm assuming all of my work is correct up until the end, but I don't know how to proceed using Trigonometric Substitution. Please help!

$\displaystyle \frac 1{\sec^2 \theta} = \cos^2 \theta = \frac 12 (1 + \cos 2 \theta)$

(and of course it's not $\displaystyle \frac 1{\tan \theta + C}$ there is no integration rule to justify such a step!)
• Jan 21st 2010, 08:52 PM
Troopa
So then...

$\displaystyle = \frac 12 (1 + \cos 2 \theta)$

$\displaystyle = \frac 12 (\theta + (cos2 \theta^2 / 2))$

Is this right?
• Jan 21st 2010, 08:54 PM
Jhevon
Quote:

Originally Posted by Troopa
So then...

$\displaystyle = \frac 12 (1 + \cos 2 \theta)$

$\displaystyle = \frac 12 (\theta + (cos2 \theta^2 / 2))$

Is this right?

no.

$\displaystyle \int\frac{1}{\sec^2\theta}d\theta=\int\cos^2\theta {d}\theta=\frac{1}{2}\int(1+\cos2\theta)d\theta=\f rac{1}{2}(\theta+\frac{1}{2}\sin{2\theta})+C$