The integral $∫\frac{dx}{\sqrt{\frac{1}{2}-5x-x^2}}$ is equal to :

$sin^{-1}\frac{2x+5}{3\sqrt{2}}+C$ $sin^{-1}\frac{2x+5}{3\sqrt{3}}+C$ $sin^{-1}\frac{2x-5}{3\sqrt{2}}+C$ $sin^{-1}\frac{2x-5}{3\sqrt{3}}+C$

$sin^{-1}\frac{2x+5}{3\sqrt{3}}+C$

The correct answer is Option (2) → $\sin^{-1}\frac{2x+5}{3\sqrt{3}}+C$ $I=∫\frac{dx}{\sqrt{\frac{1}{2}-5x-x^2}}$ $∫\frac{dx}{\sqrt{\frac{27}{4}-(x+\frac{5}{2})^2}}$ $=\sin^{-1}(\frac{x+\frac{5}{2}}{\frac{3\sqrt{3}}{2}})+C$ $=\sin^{-1}(\frac{2x+5}{3\sqrt{3}})+C$