How to find this integral?

Question

∫(x+5)/(√9-(x-3)²)

 

This problem is stumping me! Also the square root is over the whole denominator. 

Answer 1

Here is one approach, but I will leave the details to you:

 

Notice the inside of the square root has derivative -2(x - 3) = -2x + 6. A simple u-substitution will not work here, because x + 5 is not a multiple of (-2x + 6). Here is a way around this.

 

1) You can break up the numerator into (x - 3) + 8, which equals the original one.

2) Then, split the original integral into two: one with numerator (x - 3), the other with numerator 8.

3) The first integral can be done with the u-substitution u = √(9 - (x - 3²), which will work well here.

4) The second integral can be done using a trigonometric substitution x - 3 = 3 sin θ .

 

If, after trying these hints, you still need help, please post a reply.