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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 5

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.14 Question 5

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Answer:  \frac{1}{2} \log \left|x+\sqrt{x^{2}+\frac{1}{4}}\right|+c

Hint: To solve this integral, use special integral formula.

Given:  \int \frac{1}{\sqrt{1+4 x^{2}}} d x

Solution:

Let

\begin{aligned} &I=\int \frac{1}{\sqrt{1+4 x^{2}}} d x=\int \frac{1}{\sqrt{4\left(\frac{1}{4}+x^{2}\right)}} d x \\ &=\frac{1}{2} \int \frac{1}{\sqrt{x^{2}+\left(\frac{1}{2}\right)^{2}}} d x \end{aligned}

\begin{aligned} &=\frac{1}{2} \log \left|x+\sqrt{x^{2}+\left(\frac{1}{2}\right)^{2}}\right|+c \quad\quad\quad\quad\quad\quad\quad\quad\left[\because \int \frac{1}{\sqrt{x^{2}+a^{2}}} d x=\log \left|x+\sqrt{x^{2}+a^{2}}\right|+c\right] \\ \\&=\frac{1}{2} \log \left|x+\sqrt{x^{2}+\frac{1}{4}}\right|+c \end{aligned}

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