Question

Evaluate the integral
(in comments)

Answer 1

\[\int\limits_{0}^{3\sqrt3} \frac{ x^2 }{ (x^2+9)^{3/2}} dx\]

Answer 2

I know I'm suppose to use trig sub but I'm not sure which equation to use.

Answer 3

the limits of integration look like a set up for arctangent

Answer 4

So i would set a = 3 and u = x?

Answer 5

i think you would use \(x=3\tan(\theta), dx=3\sec^2(\theta)d\theta\) and go from there

Answer 6

\[\int\limits_{0}^{3\sqrt3} = \frac{ (3\tan \theta)^2 }{ (9\tan^2 \theta +9)^{3/2} }\]

Answer 7

is that set up correctly?

Answer 8

you are missing \(3\sec^2(\theta)d\theta\) up top

Answer 9

Could we simplify this expression any?
Why let x = 3tan(theta)?
What is the purpose of choosing that particular substitution?

Answer 10

I'm asking that to get us thinking about how introducing tan here might get us something that we could do something with. Are there any trig identities we could apply here?

Answer 11

\[27\int\limits_{0}^{3\sqrt3} \frac{ \tan^2\theta \sec^2 \theta }{ [9(\tan^2 \theta + 1)]^{3/2} } d \theta\]

Answer 12

Good.
And what would tan^2 + 1 equal?

Answer 13

sec^2 right?

Answer 14

Right