#### Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 74 Maths Textbook Solution.

$\sqrt{x^{2}+a x}+\frac{a}{2} \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|+c$

Hint:

To solve the given statement multiply and divide the equation by a +x.

Given:

$\int \sqrt{\frac{a+x}{x}} d x$

Solution:

$=\int \frac{a+x}{\sqrt{x(a+x)}} d x$

$=\frac{1}{2} \int \frac{2(a+x)}{\sqrt{x^{2}+a x}} d x$

$=\frac{1}{2}\left[\int \frac{2 x+a}{\sqrt{x^{2}+a x}} d x+\int \frac{a}{\sqrt{x^{2}+a x}} d x\right]$

$\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{-1}{2 \sqrt{x^{2}+a x}}(2 x+a)\right]$

$\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{a}{\sqrt{x^{2}+a x+\left(\frac{a^{2}}{4}\right)-\left(\frac{a^{2}}{4}\right)}}\right]$

$\left[d\left(\sqrt{x^{2}+a x}\right)=\frac{a}{\sqrt{\left(x+\frac{a}{2}\right)^{2}-\left(\frac{a}{2}\right)^{2}}}\right]$

$=\frac{1}{2} \cdot 2 \sqrt{x^{2}+a x}+\int \frac{a}{\sqrt{\left(x+\frac{a}{2}\right)^{2}-\left(\frac{a}{2}\right)}} d x$

$=\sqrt{x^{2}+a x}+a \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|$

$=\sqrt{x^{2}+a x}+\frac{a}{2} \log \left|\left(x+\frac{a}{2}\right)+\sqrt{x^{2}+a x}\right|+c$