First slide

Methods of integration

Question

$Evaluate \int \frac{1}{(x - 3) \sqrt{x + 1}} d x$

Easy

A

$\frac{1}{2} \log |\frac{\sqrt{x + 1} - 2}{\sqrt{x + 1} + 2}| + C$
B

$\frac{1}{2} \log |\frac{\sqrt{x + 1} + 2}{\sqrt{x + 1} - 2}| + C$
C

$\frac{1}{4} \log |\frac{\sqrt{x + 1} - 2}{\sqrt{x + 1} + 2}| + C$
D

$N o n e o f t h e s e$

Solution

$Let I = \int \frac{1}{(x - 3) \sqrt{x + 1}} d x$
$Let x + 1 = t^{2} or d x = 2 t d t$
$\begin{array}{l} ∴ I = \int \frac{1}{(t^{2} - 1 - 3)} \frac{2 t}{\sqrt{t^{2}}} d t \\ = 2 \int \frac{d t}{t^{2} - 2^{2}} = 2 \times \frac{1}{2 (2)} \log |\frac{t - 2}{t + 2}| + C \\ = \frac{1}{2} \log |\frac{\sqrt{x + 1} - 2}{\sqrt{x + 1} + 2}| + C \end{array}$

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