Question

I'm having trouble with a definite integral that involves trigonometric substitution. I was wondering if I could receive some advice on how to work it.

Answer 1

\[\int\limits_{0}^{4} (64-x^{2})^{-2}\]

Answer 2

So for this integral, you have a term which looks like: a^2-x^2
This is the form where you make a sine substitution.

Just for note, the other possible forms:
- You would use a tangent substitution for a^2+x^2
- You would use a secant substitution for x^2 - a^2

So in this case, we use sine. Draw a triangle:
So here, we can see in the drawing:\[\sin(\theta)=\frac{x}{a}\]This gives us our 'x' and 'dx':\[x = asin(\theta)\]\[dx = acos(\theta)d \theta\]
We now adapt this to the integral.

Answer 3

Sorry having connection issues

Answer 4

Okay, so what we need to do is re-draw our triangle with our specific values: