#### Need solution for RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 13 maths textbook solution.

Answer : $\inline I=-\frac{1}{2}\left(4-3 x-2 x^{2}\right)^{\frac{3}{2}}-\frac{5}{4}\left(\frac{2 x+3}{2 \sqrt{2}} \sqrt{4-3 x-2 x^{2}}+\frac{17}{4} \sin ^{-1} \frac{(2 x+3)}{\sqrt{17}}\right)+C$

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

Given : $\inline \int(3 x+1) \sqrt{4-3 x-2 x^{2}} d x$

Solution :

\inline \begin{aligned} &\text { Let, }\; \; \quad 3 x+1=a \frac{d}{d x}\left(4-3 x-2 x^{2}\right)+b \\ &\Rightarrow 3 x+1=a(-3-4 x)+b \\ &\Rightarrow 3 x+1=-4 a x+b-3 a \end{aligned}

Comparing the coefficient of x and the constant terms, we get

$4 a=-3 \Rightarrow a=-\frac{3}{4}$     and

\begin{aligned} &b-3 a=1 \Rightarrow b=1+3\left(-\frac{3}{4}\right) \Rightarrow b=-\frac{5}{4} \\ &I=\int\left[-\frac{3}{4}(-3-4 x)-\frac{5}{4}\right] \sqrt{4-3 x-2 x^{2}} d x \end{aligned}

$I=\int\left(-\frac{3}{4}\right)(-3-4 x) \sqrt{4-3 x-2 x^{2}} d x+\int-\frac{5}{4} \sqrt{4-3 x-2 x^{2}} d x$

For the first integral :

\begin{aligned} &\text { Let } 4-3 x-2 x^{2}=t \\ &\Rightarrow(-3-4 x) d x=d t \end{aligned}

$I=-\frac{3}{4} \int \sqrt{t} d t-\frac{5}{4} \int \sqrt{4+\left(\frac{3}{\sqrt{2}}\right)^{2}-\left[(\sqrt{2} x)^{2}+2 \sqrt{2} x\left(\frac{3}{\sqrt{2}}\right)+\left(\frac{3}{\sqrt{2}}\right)^{2}\right]} d x$

Use the formula : $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+C$

And $\left[\int x^{n} d x=\frac{x^{n}+1}{n+1}+C\right]$

\begin{aligned} &I=-\frac{3}{4} \frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{5}{4} \int \sqrt{\frac{17}{2}-\left(\sqrt{2} x+\frac{3}{\sqrt{2}}\right)^{2}} d x \\ &I=-\frac{3}{4} \frac{t^{\frac{3}{2}}}{\frac{3}{2}}-\frac{5}{4} \sqrt{\left(\frac{\sqrt{17}}{\sqrt{2}}\right)^{2}-\left(\sqrt{2} x+\frac{3}{\sqrt{2}}\right)^{2}} d x \end{aligned}

\begin{aligned} &I=-\frac{1}{2}\left(4-3 x-2 x^{2}\right)^{\frac{3}{2}}-\frac{5}{4}\left(\frac{\sqrt{2} x+\frac{3}{\sqrt{2}}}{2} \sqrt{4-3 x-2 x^{2}}+\frac{17}{2} \frac{1}{2} \sin ^{-1} \frac{\left(\sqrt{2} x+\frac{3}{\sqrt{2}}\right)}{\frac{\sqrt{17}}{\sqrt{2}}}\right)+C \\ &I=-\frac{1}{2}\left(4-3 x-2 x^{2}\right)^{\frac{3}{2}}-\frac{5}{4}\left(\frac{2 x+3}{2 \sqrt{2}} \sqrt{4-3 x-2 x^{2}}+\frac{17}{4} \sin ^{-1} \frac{(2 x+3)}{\sqrt{17}}\right)+C \end{aligned}