What is the integral of sqrt(9-x^2)?
What is the integral of ?
Answer & Explanation
Whenever I see these kind of functions, I recognize (by practicing a lot) that you should use a special substitution here:
You'll see why we're making this odd substitution, despite the fact that it may seem strange at first.
Replace everyhting in the integral:
We can bring the 3 out of the integral:
You can factor the 9 out:
We know the identity:
If we solve for , we get:
This is exactly what we see in the integral, so we can replace it:
You might know this one as a basic antiderivative, but if you don't, you can figure it out like so:
We use the identity:
(you can work this out by substitution)
Now, all we have to do is put into the function. Let's look back at how we defined it:
To get out of this, you need to take the inverse function of on both sides, this is :
Now we need to insert it into our solution: