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#### Please Solve RD Sharma Class 12 Chapter Indefinite Integrals Exercise 18.17 Question 9 Maths Textbook Solution

Answer:- $\frac{1}{\sqrt{5}}\log \left | \left ( x-\frac{1}{5} \right )+\sqrt{x^{2}-\frac{2x}{5}} \right |+c$

Hint: - To solve this problem, use special integration formula

Given:- $\int \frac{1}{\sqrt{5x^{2}-2x}}dx$

Solution:-

$Let\: I=\int \frac{1}{\sqrt{5x^{2}-2x}}dx=\int \frac{1}{\sqrt{5\left ( x^{2}-\frac{2}{5}x \right )}}dx$

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

\begin{aligned} &\text { put } \mathrm{x}-\frac{1}{5}=t \Rightarrow d x=d t \text { then } \\ &\mathrm{I}=\frac{1}{\sqrt{5}} \int \frac{1}{\sqrt{t^{2}-\left(\frac{1}{5}\right)^{2}}} d t \\ &I=\frac{1}{\sqrt{5}} \log \left|t+\sqrt{t^{2}-\left(\frac{1}{5}\right)^{2}}\right|+c \quad\quad\quad\quad\quad\quad\left[\because \int \frac{1}{\sqrt{x^{2}-a^{2}}} d x=\log \mid x+\sqrt{x^{2}-a|}+c\right] \end{aligned}
\begin{aligned} &I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{\left(x-\frac{1}{5}\right)^{2}-\left(\frac{1}{5}\right)^{2}}\right|+c \quad\quad\quad\quad\left(\because t=x-\frac{1}{5}\right) \\ &\Rightarrow I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{x^{2}-\frac{2 x}{5}+\frac{1}{25}-} \frac{1}{25}\right|+c \\ &\Rightarrow I=\frac{1}{\sqrt{5}} \log \left|\left(x-\frac{1}{5}\right)+\sqrt{x^{2}-\frac{2 x}{5}}\right|+c \end{aligned}