# Thread: Annoying trigonometric substitution integral

1. ## Annoying trigonometric substitution integral

How do you solve this integral: From 0 to a, x^2*sqrt(a^2-x^2)

I have no idea where pi comes from, so here's what I do

Let x=a*sin(theta)

My work ends up to be (4a^4/32) - (a^4sin4a/32)

Help!

2. ## Re: Annoying trigonometric substitution integral

When you get your new integral in the theta domain, what is the expression for that?

3. ## Re: Annoying trigonometric substitution integral

Originally Posted by some_nerdy_guy
How do you solve this integral: From 0 to a, x^2*sqrt(a^2-x^2)

I have no idea where pi comes from, so here's what I do

Let x=a*sin(theta)
Unfortunately, you error occurs somewhere in here- in exactly the part you did not show. Did you change the limits of integration as you worked? If x= a sin(theta), then when x= 0, a sin(theta)= 0 so theta= 0. When x= a, a= a sin(theta) so sin(theta)= 1, theta= pi/2.

My work ends up to be (4a^4/32) - (a^4sin4a/32)

Help!

4. ## Re: Annoying trigonometric substitution integral

Okay, thanks guys. I've figured out the problem, which is, as HallsofIvy stated, I didn't change my limits of integration in which 'a' should have been 'pi/2.' Could anyone possibly explain why if x=a, then theta=pi/2. I don't understand the logic behind that.

5. ## Re: Annoying trigonometric substitution integral

Originally Posted by some_nerdy_guy
Could anyone possibly explain why if x=a, then theta=pi/2. I don't understand the logic behind that.
That's because of the substitution, the substitution was $x=a\sin(\theta) \Leftrightarrow \theta=\arcsin\left(\frac{x}{a}\right)$
So if $x=a$ that means $\theta=\arcsin\left(\frac{a}{a}\right)=\arcsin(1)= \frac{\pi}{2}$

6. ## Re: Annoying trigonometric substitution integral

Ahhh, okay. I really need to brush up on my trig.

7. ## Re: Annoying trigonometric substitution integral

When you do a substitution on a definite integral, you must change over three things: the limits, the integrand, and the differential. For an indefinite integral, of course, while you don't need the limits, you must still transform the integrand and the differential.