First slide

Methods of integration

Question

$Evaluate \int \sqrt{\frac{1 + x}{x}} d x$

Moderate

A

$\sqrt{x^{2} + x} + \log |(x + \frac{1}{2}) + \sqrt{x^{2} + x}| + C$
B

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

$\sqrt{x^{2} + x} + \frac{1}{2} \log |(x + \frac{1}{2}) + \sqrt{x^{2} + x}| + C$
D

None of these

Solution

$\begin{array}{l} \int \sqrt{\frac{1 + x}{x}} d x \\ = \int \sqrt{\frac{1 + x}{x}} \sqrt{\frac{1 + x}{1 + x}} d x \\ = \int \frac{1 + x}{\sqrt{x^{2} + x}} d x \\ = \frac{1}{2} \int \frac{2 x + 1}{\sqrt{x^{2} + x}} d x + \frac{1}{2} \int \frac{1}{\sqrt{x^{2} + x}} d x \\ = \frac{1}{2} \int \frac{1}{\sqrt{t}} d t + \frac{1}{2} \int \frac{1}{\sqrt{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}}} d x, where t = x^{2} + x \\ = \sqrt{t} + \frac{1}{2} \log |(x + \frac{1}{2}) + \sqrt{x^{2} + x}| + C \\ = \sqrt{x^{2} + x} + \frac{1}{2} \log |(x + \frac{1}{2}) + \sqrt{x^{2} + x}| + C \end{array}$

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