Integration with fraction and square root

[Dell]

i am given this function and need to integrate it

$$\frac{4x+7}{\sqrt{-4x^2+20x-9}}$$

i have been trying to intergrate it by calling something T, preferably the expression under the sqrd root, or part thereof (-4x²+20x=T) but i can't find the way to do this, can't find the best expression that fits both the denominator and numerator of the fraction.. any other ideas??

 

[Mark44]

Try u = -4x^2 + 20x - 9, and du = (-8x + 20)dx (I like u for substitutions better than T.)

Your numerator is 4x + 7 = 4x -10 + 17, so your integral can be rewritten as two integrals.

$$\int \frac{4x - 10}{\sqrt{-4x^2 + 20x - 9}}dx + \int \frac{17}{\sqrt{-4x^2 + 20x - 9}}dx$$

Applying the substitution in the first integral, we have $-1/2\int du/u^{1/2}$. The second one is probably amenable to a trig substitution.