#### Please solve RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 2 maths textbook solution.

Answer : $\frac{1}{6}\left(2 x^{2}+3\right)^{\frac{3}{2}}+\frac{x}{2} \sqrt{x^{2}-x+1}+\frac{3 \sqrt{2}}{4} \log \left|\frac{\sqrt{2 x}+\sqrt{2 x^{2}+3}}{\sqrt{3}}\right|+C$

Hint: To solve the given integration, we express the linear term as a derivative of quadratic into constant plus another constant

Given : $\int(x+1) \sqrt{2 x^{2}+3} d x$

Solution : $\text { Let } I=\int x \sqrt{2 x^{2}+3} d x+\int \sqrt{2 x^{2}+3} d x$

\begin{aligned} &I_{1}=\int x \sqrt{2 x^{2}+3} d x \\ &\text { Let } 2 x^{2}+3=t \\ &4 x d x=d t \\ &x \cdot \mathrm{d} x=\frac{d t}{4} \end{aligned}

\begin{aligned} &I_{1}=\int \sqrt{t} \frac{d t}{4} \\ &=\frac{1}{4} \int t^{\frac{1}{2}} d x \end{aligned}

$=\frac{1}{4} \frac{t^{1 / 2+1}}{1 / 2+1} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int x^{n} d x=\frac{x^{n}+1}{n+1}\right]$

\begin{aligned} &=\frac{1}{4} \times 2 \frac{t^{\frac{3}{2}}}{3} \\ &=\frac{t^{\frac{3}{2}}}{6}=\frac{\left(2 x^{2}+3\right)^{\frac{3}{2}}}{6}+C_{1} \end{aligned}                                           ....(i)

$I_{2}=\int \sqrt{2 x^{2}+3} d x$

Use the formula : $\int \sqrt{x^{2}+a^{2}} d x=\left[\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+C\right]$

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

$I_{2}=\sqrt{2}\left[\frac{x}{2} \sqrt{x^{2}+\frac{3}{2}}+\frac{3}{2 \times 2} \log \left|x+\sqrt{x^{2}+\frac{3}{2}}\right|\right]+C_{2}$                               ....(ii)

Adding (i) and (ii) ; $I=I_{1}+I_{2}$

$I=\int x \sqrt{2 x^{2}+3} d x+\int \sqrt{2 x^{2}+3} d x$

\begin{aligned} &=\frac{\left(2 x^{2}+3\right)^{\frac{3}{2}}}{6}+\frac{x}{\sqrt{2}} \sqrt{x^{2}+\frac{3}{2}}+\frac{3}{2 \sqrt{2}} \log \left|x+\sqrt{x^{2}+\frac{3}{2}}\right|+C \\ &=\frac{1}{6}\left(2 x^{2}+3\right)^{\frac{3}{2}}+\frac{x}{\sqrt{2}} \frac{\sqrt{2 x^{2}+3}}{\sqrt{2}}+\frac{3}{2 \sqrt{2}} \log \left|x+\frac{\sqrt{2 x^{2}+3}}{\sqrt{2}}\right|+C \end{aligned}

\begin{aligned} &=\frac{1}{6}\left(2 x^{2}+3\right)^{\frac{3}{2}}+\frac{x}{2} \sqrt{2 x^{2}+3}+\frac{3}{2 \sqrt{2}} \frac{\sqrt{2}}{\sqrt{2}} \log \left|\frac{\sqrt{2} x+\sqrt{2 x^{2}+3}}{\sqrt{2}}\right|+C \\ &\int(x+1) \sqrt{2 x^{2}+3} d x=\frac{1}{6}\left(2 x^{2}+3\right)^{\frac{3}{2}}+\frac{x}{2} \sqrt{2 x^{2}+3}+\frac{3 \sqrt{2}}{4} \log \left|\frac{\sqrt{2} x+\sqrt{2 x^{2}+3}}{\sqrt{2}}\right|+c \end{aligned}