#### Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 55 Maths Textbook Solution.

$I=\frac{2}{a^{\frac{3}{2}}}\left[\frac{(a-2)(1-\sqrt{a x})}{2}+\frac{(1-\sqrt{a x})^{2}}{2}+(1-a) \ln |(1-\sqrt{a x})|\right]+c$

Given:

$\int \frac{\sqrt{a}-\sqrt{x}}{1-\sqrt{a x}} d x$

Hint:

To solve this equation we have to  suppose denominator as t.

Solution:

$I=\int \frac{\sqrt{a}-\sqrt{x}}{1-\sqrt{a x}} d x$

$1-\sqrt{a x}=t$

$-\sqrt{a x}=t-1$

$-\frac{a}{2 \sqrt{a x}} d x=d t$

$d x=\frac{-2 \sqrt{a x}}{a} d t$

$d x=\frac{2(t-1)}{a} d t$

$I=\frac{1}{\sqrt{a}} \int \frac{a-\sqrt{a x}}{1-\sqrt{a x}} d x$

$I=\frac{1}{\sqrt{a}} \int \frac{a+(t-1)}{t}\left(\frac{2(t-1)}{a}\right) d t$

$I=\frac{2}{a^{\frac{3}{2}}} \int \frac{(t-1)(a+t-1)}{t} d t$

$I=\frac{2}{a^{\frac{3}{2}}} \int \frac{a t+t^{2}-t-a-t+1}{t} d t$

$I=\frac{2}{a^{\frac{3}{2}}} \frac{t(a-2)+t^{2}+(1-a)}{t} d t$

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

$I=\frac{2}{a^{\frac{3}{2}}}\left[(a-2) t+\frac{t^{2}}{2}+(1-a) \ln |t|\right]+c$

$I=\frac{2}{a^{\frac{3}{2}}}\left[\frac{(a-2)(1-\sqrt{a x})}{2}+\frac{(1-\sqrt{a x})^{2}}{2}+(1-a) \ln |(1-\sqrt{a x})|\right]+c$