Calculate the value for the following improper integral

Question 1: your way is not correct (reason: see my answer to question 2).

Question 2: $\int_0^{\infty}$ is convergent $ \iff$ the integrals $\int_0^{1},\int_{1}^{3/2}$ and $\int_{3/2}^{\infty} $ are all convergent.

Now show that $\int_0^{1}\frac{dx}{2x^2-5x+3}$ is divergent !

Conclusion: $\int_0^{\infty}\frac{dx}{2x^2-5x+3}$ is divergent.