Solve Difficult Integral: Help Appreciated

[dats13]

I have tried many different ways of solving this integral but always seem to get stuck. Any help on this one would be greatly appreciated.

 

[ideasrule]

Have you tried completing the square for the thing under the square root sign?

 

[w3390]

You could also multiply the stuff under the radical out and use u substitution to solve.

 

[dats13]

I have tried completing the square. I'll check it again. I'll also try a substitution. Thanks

 

[w3390]

Actually, sorry, I don't think u substitution works here.

 

[Mark44]

If you multiply the factors in the radical you get -x^2 + 9x -18. If the numerator were -2x + 9, you would have $\int u^{-1/2}du$.

The solution is to add what you need in the numerator, and then subtract it off, and split into two separate integrals. This is slightly more complicated in that you need a multiplier of -2 for x.

After splitting into two integrals, the first integral can be done by an ordinary substitution, as described above. The second can be done by completing the square in the radical, and using a trig substitution.

Caveat: I haven't done this problem, but I think this strategy will work.

 

[dats13]

Thanks for all of intput. I did end up sovling this by spliting the integral into two. The first part was a radical and the second turned out to be an arcsin.