simple integral help needed work shown

johnq2k7

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Feb 10, 2009
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46
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
 
xx2+6x8dx\displaystyle \int x\sqrt{-x^{2}+6x-8}dx

Make the sub u=x3,   u+3=x,   dx=du\displaystyle u=x-3, \;\ u+3=x, \;\ dx=du

Then, we get 1u2(u+3)du\displaystyle \int \sqrt{1-u^{2}}(u+3)du

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

sin(w)dw+3dwsin3(w)dw3sin2(w)dw\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.
 
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