First slide

Introduction to integration

Question

$\int \frac{dx}{\sqrt{1 + 4 x^{2}}}$ is equal to

Easy

A

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

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

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

None of these

Solution

$\int \frac{dx}{\sqrt{1 + 4 x^{2}}} = \int \frac{dx}{\sqrt{4 \{{(\frac{1}{2})}^{2} + x^{2}\}}}$
$\begin{array}{r} = \frac{1}{2} \int \frac{dx}{\sqrt{x^{2} + {(\frac{1}{2})}^{2}}} = \frac{1}{2} | \log | x + {\sqrt{x^{2} + (\frac{1}{2})}}^{2} ∣ + c \\ [∵ \int \frac{1}{\sqrt{x^{2} + a^{2}}} dx = \log |x + \sqrt{x^{2} + a^{2}}|] \end{array}$
$\begin{aligned} = \frac{1}{2} \log |x + \frac{\sqrt{4 x^{2} + 1}}{2}| + C \\ = \frac{1}{2} \log |2 x + \sqrt{4 x^{2} + 1}| - \frac{1}{2} \log 2 + C \\ = [∵ \log (\frac{m}{n}) = \log m - \log n] \\ = \frac{1}{2} \log |2 x + \sqrt{4 x^{2} + 1}| + C \end{aligned}$
$[: \frac{1}{2} \log 2 is constant and constant + constant = C]$

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