1. ## Integral

Hello!

$\displaystyle \int{\frac{xdx}{(x^2-3x+2)\sqrt{x^2-4x+3}}}$

Haven't been much familiar with these, so I need some help!
I think there's a hint but but I'm not sure if it's right: $\displaystyle \sqrt{(x-1)(x-3)}=t(x-3)$, anyway I don't know how to use it!

Thanks

2. Alright, then make the substitution $\displaystyle t=\frac{\sqrt{(x-1)(x-3)}}{x-3}=\sqrt{\frac{x-1}{x-3}}$. Solve for x, calculate $\displaystyle dx$ in terms of $\displaystyle dt$, then substitute everything in the integral in terms of t. When I do that I get $\displaystyle \int \frac{3t^2-1}{t^2+t^4}dt$. Integrate, then bring everything back to x.

3. Ok thanks. One more question, Is there any order how to make these substitutions or we just have to be vigilant?

Thanks

4. You may have to experiment abit. Try a substitution here and there, and if it looks like a deadend to you, try another one. With practice, you will learn to see what substitutions to make on the go. Sometimes you may need to use integration by parts rather than substitution.