Science:Math Exam Resources/Courses/MATH101/April 2017/Question 01 (b)/Solution 1

Consider the substitution $u = x^{2} - 3$ . Then $d u = 2 x d x$ because the derivative of $x^{2} - 3$ with respect to $x$ is $2 x$ . Hence,

$\int \frac{3 x}{\sqrt{x^{2} - 3}} d x = \frac{3}{2} \int \frac{2 x d x}{\sqrt{x^{2} - 3}} = \frac{3}{2} \int \frac{d u}{\sqrt{u}}$

By the Power Rule for the integration, we get

$\int \frac{d u}{\sqrt{u}} = \int u^{- \frac{1}{2}} d u = \frac{1}{- \frac{1}{2} + 1} u^{- \frac{1}{2} + 1} + C = 2 \sqrt{u} + C$

where $C$ is an arbitrary constant (the constant of integration)

So,

$\int \frac{3 x}{\sqrt{x^{2} - 3}} d x = \frac{3}{2} \int \frac{d u}{\sqrt{u}} = \frac{3}{2} (2 \sqrt{u} + C)$

and substituting $x^{2} - 3$ for $u$ into the above expression now gives

$\int \frac{3 x}{\sqrt{x^{2} - 3}} d x = 3 \sqrt{u} + C = 3 \sqrt{x^{2} - 3} + C$