How can I evaluate this integral using basic integration techniques?

[whatlifeforme]

Homework Statement

I am reviewing some basic integration techniques. how would i evaluate this?

Homework Equations

$\displaystyle\int {\frac{1}{\sqrt{8x-x^2}} dx}$

The Attempt at a Solution

i've tried multiplying by

$\frac{\sqrt{8x-x^2}}{\sqrt{8x-x^2}}$ but that isn't getting me anywhere.

should i use something like completing the square?

 

[mfb]

Completing the square is a good way to begin. Afterwards, a clever substitution can help.

 

[whatlifeforme]

okay. I am stuck with the subtracted term.

$\frac{1}{-\sqrt{(x^2 - 8x + 16) - 16}}$

$\frac{1}{-\sqrt{(x-4)^2 - 16}}$

 

[whatlifeforme]

i don't think completing the square works here.

 

[eumyang]

okay. I am stuck with the subtracted term.

$\frac{1}{-\sqrt{(x^2 - 8x + 16) - 16}}$

$\frac{1}{-\sqrt{(x-4)^2 - 16}}$

Looks like you pulled a negative outside of the square root. You can't do that! Try again.

 

[phosgene]

Once you've fixed up the negative, try the substitution u=(x-4) and then consider what the derivatives of the inverse trigonometric functions look like.

 

[whatlifeforme]

great. thanks phosgene.

so we are looking at:

answer:
$arcsin(\frac{x-4}{4}) + c$