Answer
$$\int\frac{dx}{x\sqrt{7+x^2}}=-\frac{1}{\sqrt7}\ln\Big|\frac{\sqrt7+\sqrt{7+x^2}}{x}\Big|+C$$
Work Step by Step
$$A=\int\frac{dx}{x\sqrt{7+x^2}}$$
Use Formula 40, which states that $$\int \frac{dx}{x\sqrt{a^2+x^2}}=-\frac{1}{a}\ln\Big|\frac{a+\sqrt{a^2+x^2}}{x}\Big|+C$$
for $a=\sqrt7$ here:
$$A=-\frac{1}{\sqrt7}\ln\Big|\frac{\sqrt7+\sqrt{7+x^2}}{x}\Big|+C$$
