Can this algebraic integral be solved using trigonometric substitutions?

[utkarshakash]

Homework Statement

$\int \dfrac{x+2}{\sqrt{(x-2)(x-3)}} dx$

The Attempt at a Solution

I've tried substitutions like assuming (x-2) = t^2 or x= 1/t or x=1/t^2, but none of them seems to ease the problem. Breaking the integral into two helps to integrate the second but first integral still remains complicated. I'm also sure that trig substitutions won't work here.

 

[Tanya Sharma]

Homework Statement

$\int \dfrac{x+2}{\sqrt{(x-2)(x-3)}} dx$

The Attempt at a Solution

I've tried substitutions like assuming (x-2) = t^2 or x= 1/t or x=1/t^2, but none of them seems to ease the problem. Breaking the integral into two helps to integrate the second but first integral still remains complicated. I'm also sure that trig substitutions won't work here.

Rewrite the given integral as $\frac{1}{2} \int \dfrac{2x-5}{\sqrt{x^2-5x+6}} dx$ + $\frac{9}{2}\int \dfrac{1}{\sqrt{(x-\frac{5}{2})^2-(\frac{1}{2})^2}} dx$

I hope the two integrals are easy to handle .