What is the integral of sqrt(9-x^2)?

Gingan7mhd

Gingan7mhd

Answered question

2023-03-25

What is the integral of 9 - x 2 ?

Answer & Explanation

Ryker Lloyd

Ryker Lloyd

Beginner2023-03-26Added 6 answers

Whenever I see these kind of functions, I recognize (by practicing a lot) that you should use a special substitution here:
9 - x 2 d x
x = 3 sin ( u )
You'll see why we're making this odd substitution, despite the fact that it may seem strange at first.
d x = 3 cos ( u ) d u
Replace everyhting in the integral:
9 - ( 3 sin ( u ) ) 2 3 cos ( u ) d u
We can bring the 3 out of the integral:
3 9 - ( 3 sin ( u ) ) 2 cos ( u ) d u
3 9 - 9 sin 2 ( u ) cos ( u ) d u
You can factor the 9 out:
3 9 ( 1 - sin 2 ( u ) ) cos ( u ) d u
3 3 1 - sin 2 ( u ) cos ( u ) d u
We know the identity: cos 2 x + sin 2 x = 1
If we solve for cos x , we get:
cos 2 x = 1 - sin 2 x
cos x = 1 - sin 2 x
This is exactly what we see in the integral, so we can replace it:
9 cos 2 ( u ) d u
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: cos 2 ( u ) = 1 + cos ( 2 u ) 2
9 1 + cos ( 2 u ) 2 d u
9 2 1 + cos ( 2 u ) d u
9 2 ( 1 d u + cos ( 2 u ) d u )
9 2 ( u + 1 2 sin ( 2 u ) ) + C (you can work this out by substitution)
9 2 u + 9 4 sin ( 2 u ) + C
Now, all we have to do is put u into the function. Let's look back at how we defined it:
x = 3 sin ( u )
x 3 = sin ( u )
To get u out of this, you need to take the inverse function of sin on both sides, this is arcsin :
arcsin ( x 3 ) = arcsin ( sin ( u ) )
arcsin ( x 3 ) = u
Now we need to insert it into our solution:
9 2 arcsin ( x 3 ) + 9 4 sin ( 2 arcsin ( x 3 ) ) + C

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