#### Provide solution for RD Sharma maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.29 Question 6 maths textbook solution.

Answer : $\inline \\I=\frac{1}{6}\left(2 x^{2}-6 x+5\right)^{3 / 2}-\frac{1}{2}\left[\frac{2 x-3}{2} \sqrt{x^{2}-3 x+\frac{5}{2}}+\frac{11}{8} \log \left|\frac{2 x-3}{\sqrt{2}}+\sqrt{2 x^{2}-6 x+5}\right|\right]+C$

Hint: We solve this integration by qualitative derivation.

Given: $\int(x-2) \sqrt{2 x^{2}-6 x+5} \; d x$

Let, $x-2=a \frac{d}{d x}\left(2 x^{2}-6 x+5\right)+b$

\begin{aligned} &\Rightarrow x-2=a(4 x-6)+b \\ &\Rightarrow x-2=4 a x+b-6 a \end{aligned}

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

\begin{aligned} &4 a=1 \Rightarrow a=\frac{1}{4} \text { and } \\ &b-6 a=-2 \Rightarrow b=6\left(\frac{1}{4}\right)-2 \Rightarrow b=\left(-\frac{1}{2}\right) \\ &I=\int\left[\frac{1}{4}(4 x-6)+\left(-\frac{1}{2}\right)\right] \sqrt{2 x^{2}-6 x+5} d x \end{aligned}

$I=\int \frac{1}{4}(4 x-6) \sqrt{2 x^{2}-6 x+5} d x+\int-\frac{1}{2} \sqrt{2 x^{2}-6 x+5} d x$

For the first integral : Let, $2 x^{2}-6 x+5=t \Rightarrow(4 x-6) d x=d t$

$I=\frac{1}{4} \int \sqrt{t} d t+\left(-\frac{1}{2}\right) \int \sqrt{(\sqrt{2} x)^{2}-2(\sqrt{2} x)\left(\frac{3}{\sqrt{2}}\right)+\left(\frac{3}{\sqrt{2}}\right)^{2}+5-\frac{9}{4}} d x$

\begin{aligned} &I=\frac{1}{4} \frac{t^{\frac{1}{2}+1}}{3 / 2}-\frac{1}{2} \int \sqrt{\left(\sqrt{2} x-\frac{3}{\sqrt{2}}\right)^{2}+\frac{11}{4}} d x \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int x^{n} d x=\frac{x^{n}+1}{n+1}+C\right] \\ &I=\frac{1}{2 \times 3} t^{3 / 2}-\frac{1}{2} \int \sqrt{\left(\sqrt{2} x-\frac{3}{\sqrt{2}}\right)^{2}+\left(\frac{\sqrt{11}}{2}\right)^{2}} d x \end{aligned}

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

\begin{aligned} &I=\frac{1}{6}\left(2 x^{2}-6 x+5\right)^{\frac{3}{2}}-\frac{1}{2}\left[\frac{\left(\sqrt{2} x-\frac{3}{\sqrt{2}}\right)}{2} \sqrt{2 x^{2}-6 x+5}+\frac{\frac{11}{4}}{2} \log \left|\left(\sqrt{2} x-\frac{3}{\sqrt{2}}\right)+\sqrt{2 x^{2}-6 x+5}\right|\right]+C \\ &I=\frac{1}{6}\left(2 x^{2}-6 x+5\right)^{3 / 2}-\frac{1}{2}\left[\frac{2 x-3}{2} \sqrt{x^{2}-3 x+\frac{5}{2}}+\frac{11}{8} \log \left|\frac{2 x-3}{\sqrt{2}}+\sqrt{2 x^{2}-6 x+5}\right|\right]+C \end{aligned}