Indefinite integration of $1/\sqrt{3-5x-2x^2}$

Question

Cannot make it out.

$$\int \frac{dx}{(3-5x-2x^2)^{1/2}} $$

Is the problem correct, or does it have errors? I have a doubt.

Answer 1

This function is integrable in elementary terms.

$$\int\frac1{\sqrt{3-5x-2x^2}}\,dx=\frac1{\sqrt2}\cdot\arcsin\left(\dfrac{4x+5}7\right)+C$$

It can be done with a combination of complete the square and rewriting the integral into an integral definition of $\arcsin x$. $$\int\frac1{\sqrt{1-x^2}}\,dx=\arcsin x+ C$$