Question

Evaluate the integral \(\int {\frac{x}{{\sqrt {{x^2} - 7} }}dx} \)

Short answer

The Evaluation of Equation Gives us,\(\int {\frac{x}{{\sqrt {{x^2} - 7} }}dx} = \sqrt {{x^2} - 7} + c\)

Step-by-step solution

Find the integral.

\(\int {\frac{x}{{\sqrt {{x^2} - 7} }}dx} \)

Let

\(\begin{aligned}{l}{x^2} - 7 &= t\\2x\,\,dx &= dt\\dx &= \frac{{dt}}{{2x}}\end{aligned}\)

\(\begin{aligned}{l}\int {\frac{x}{{\sqrt {{x^2} - 7} }}dx} &= \int {\frac{x}{{\sqrt t }}} \frac{{dt}}{{2x}}\\ &= \frac{1}{2}\int {\frac{1}{t}dt} \\ &= \frac{1}{2}\int {{t^{ - \frac{1}{2}}}} dt\\ &= \frac{1}{2} \times \frac{{{t^{\frac{1}{2}}}}}{{\frac{1}{2}}} + c\\ &= {t^{\frac{1}{2}}} + c\\ &= \sqrt {{x^2} - 7} + c\end{aligned}\)

Hence, \(\int {\frac{x}{{\sqrt {{x^2} - 7} }}dx} = \sqrt {{x^2} - 7} + c\)