#### Please solve RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 5 maths textbook solution.

Answer: $\inline \\\frac{4}{3}\left(x^{2}+x+1\right)^{\frac{3}{2}}+\frac{3(2 x+1)}{8} \sqrt{x^{2}+x+1}+\frac{27}{8} \log \left|\left(x+\frac{1}{2}\right)+\sqrt{x^{2}+x+1}\right|+C$

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

Given:   $\inline I=\int(4 x+1) \sqrt{x^{2}-x-2} d x$

Solution: Let, $\inline 4 x+1=M \frac{d}{d x}\left(x^{2}-x-2\right)+N$

\inline \begin{aligned} &\Rightarrow 4 x+1=M(2 x-1)+N \\ &\Rightarrow 4 x+1=2 M x-M+N \end{aligned}

Now comparing the coefficients of x and the constant term, we get

\inline \begin{aligned} &2 M=4 \Rightarrow M=2 \text { and }-M+N=1 \Rightarrow-2+N=1 \Rightarrow N=3 \\ &I=2 \int(2 x-1) \sqrt{x^{2}-x-2} d x+3 \int \sqrt{x^{2}-x-2} d x \end{aligned}

For first integral, let, $\inline x^{2}-x-2=t \Rightarrow(2 x-1) d x=d t$

For second integral, $\inline x^{2}-x-2=x^{2}-2 \cdot x \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}-2=\left(x-\frac{1}{2}\right)^{2}-\frac{9}{4}=\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}$

So, the integral becomes

$\inline I=2 \int \sqrt{t} d t+3 \int \sqrt{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}} d x$

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

\begin{aligned} &I=2 \frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+3\left[\frac{\left(x-\frac{1}{2}\right)}{2} \sqrt{x^{2}-x-2}-\frac{\frac{9}{4}}{2} \log \left|\left(x-\frac{1}{2}\right)+\sqrt{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}\right|\right]+C \\ &I=2 \times 2 \frac{t^{\frac{3}{2}}}{3}+\frac{3}{2}\left[\left(x-\frac{1}{2}\right) \sqrt{x^{2}-x-2}-\frac{9}{4} \log \left|\left(x-\frac{1}{2}\right)+\sqrt{x^{2}-x-2}\right|\right]+C \end{aligned}

$\\I=\frac{4}{3}\left(x^{2}-x-2\right)^{3 / 2}+\frac{3}{4}(2 x-1)\sqrt{x^{2} x-2}-\frac{27}{8}\log \left|\left(x-\frac{1}{2}\right)+\sqrt{x^{2}-x-2}\right|+C$