What is the integral of #sqrt(9-x^2)#?
Whenever I see these kind of functions, I recognize (by practicing a lot) that you should use a special substitution here:
This might look like a weird substitution, but you're going to see why we're doing this.
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
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:
Now, all we have to do is put
Now we need to insert it into our solution:
This is the final solution.
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