#### Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.28 Question 2

$\left(\frac{2 x+1}{4}\right) \sqrt{x^{2}+x+1}+\frac{3}{8} \log \left|(2 x+1)+\sqrt{x^{2}+x+1}\right|+c$

Hint:-

$\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+c$

Given:-

$\int \sqrt{x^{2}+x+1} d x$

Solution:-

\begin{aligned} &\int \sqrt{x^{2}+x+1} d x \\\\\ &=\int \sqrt{x^{2}+x+\frac{1}{4}+\frac{3}{4}} d x \\\\ &=\int \sqrt{x^{2}+x+\left(\frac{1}{5}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}} d x \end{aligned}

\begin{aligned} &=\int \sqrt{\left(x+\frac{1}{5}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}} d x \\\\ &=\frac{\left(x+\frac{1}{5}\right)}{2} \sqrt{\left(x+\frac{1}{5}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}+\frac{\left(\frac{\sqrt{3}}{2}\right)^{2}}{2} \log \left|\left(x+\frac{1}{5}\right)+\sqrt{\left(x+\frac{1}{5}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}\right|+c \end{aligned}

Using the formula

\begin{aligned} &\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+c \\\\ &=\left(\frac{2 x+1}{4}\right) \sqrt{x^{2}+x+1}+\frac{3}{8} \log \left|\left(x+\frac{1}{5}\right)+\sqrt{\left(x+\frac{1}{5}\right)^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}}\right|+c \end{aligned}

\begin{aligned} &=\left(\frac{2 x+1}{4}\right) \sqrt{x^{2}+x+1}+\frac{3}{8} \log \left|\left(\frac{2 x+1}{2}\right)+\frac{1}{2} \sqrt{x^{2}+x+1}\right|+c \\\\ &=\left(\frac{2 x+1}{4}\right) \sqrt{x^{2}+x+1}+\frac{3}{8} \log \left|(2 x+1)+\sqrt{x^{2}+x+1}\right|+c \end{aligned}