evaluate ∫ x / √(x² - 7) dx.

Explanation:

Step1: Use substitution

Let $u = x^{2}-7$, then $du = 2x dx$, and $x dx=\frac{1}{2}du$.

Step2: Rewrite the integral

The integral $\int\frac{x}{\sqrt{x^{2}-7}}dx$ becomes $\frac{1}{2}\int\frac{du}{\sqrt{u}}$.

Step3: Integrate

We know that $\int u^{n}du=\frac{u^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $n=-\frac{1}{2}$, $\frac{1}{2}\int u^{-\frac{1}{2}}du=\frac{1}{2}\times\frac{u^{\frac{1}{2}}}{\frac{1}{2}}+C$.

Step4: Substitute back

Substitute $u = x^{2}-7$ back, we get $\sqrt{x^{2}-7}+C$.

Answer:

$\sqrt{x^{2}-7}+C$