simple integral help needed work shown

[johnq2k7]

Solve the integral of (x)(sqrt(6x-x^2-8) dx:

work shown:

-x^2+6*x-8= -(x^2-6x-8)
= - (x^2-6x+9)+1
= -(x-3)^2 +1

let u=x
du= dx

therefore, integral (x)(sqrt(6x-x^2-8)= u*(sqrt(1-(u-3)^2))

using IBP: I keep getting a continous stream of integration with a solution as I continue to do IBP.. is there another effective method I'm not using please help out

 

[galactus]

\(\displaystyle \int x\sqrt{-x^{2}+6x-8}dx\)

Make the sub \(\displaystyle u=x-3, \;\ u+3=x, \;\ dx=du\)

Then, we get \(\displaystyle \int \sqrt{1-u^{2}}(u+3)du\)

Now, a little trig sub, \(\displaystyle u=sin(w), \;\ du=cos(w)dw\) gives:

\(\displaystyle \int sin(w)dw+\int 3 dw-\int sin^{3}(w)dw-3\int sin^{2}(w)dw\)

These can easily be done individually and then back sub back to x.