Does the integral $\displaystyle \int_{1}^{\infty} \frac{3 \; dx}{2x \sqrt{4+5x}} $ converge or diverge?
If it converges, what is its value?
$\displaystyle \int_{1}^{\infty} \frac{3x}{2x \sqrt{4+5x}}~dx = \frac 32 \lim_{N \to \infty} \int_{1}^{N} \frac 1{\sqrt {4 + 5x}}~dx$
We proceed by substitution:
Let $\displaystyle u = 4 + 5x$
$\displaystyle \Rightarrow du = 5 ~dx$
$\displaystyle \Rightarrow \frac 15~du = dx$
So our integral becomes:
$\displaystyle \frac 3{10} \lim_{N \to \infty} \int_{x = 1}^{x = N} u^{- \frac 12}~du = \frac 3{10} \left[ 2 u^{\frac 12} \right]_{x = 1}^{x = N}$
$\displaystyle = \frac 3{10} \lim_{N \to \infty} \left[ 2 \sqrt {4 + 5x} \right]_{1}^{N}$
I leave the rest to you