First slide

Methods of integration

Question

$\int \frac{1}{x - \sqrt{x}} dx$ is equal to

Easy

A

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

$\log | \sqrt{x} + 1 | + C$
C

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

$2 \log | x + 1 | + C$

Solution

$\begin{array}{l} \int \frac{1}{x - \sqrt{x}} dx = \int \frac{1}{\sqrt{x} (\sqrt{x} - 1)} dx \\ Let \sqrt{x} - 1 = t \Rightarrow \frac{1}{2 \sqrt{x}} = \frac{dt}{dx} \Rightarrow dx = 2 \sqrt{x} dt \\ ∴ \begin{matrix} \frac{1}{\sqrt{x} (\sqrt{x} - 1)} dx & = \int \frac{1}{\sqrt{x} t} 2 \sqrt{x} dt = \int \frac{2}{t} dt = 2 \cdot \log | t | + C \\ = 2 \log | \sqrt{x} - 1 | + C \end{matrix} \end{array}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App