Question

Integrate :- dx /√x√5x+7

\(\int\frac{dx}{\sqrt x\sqrt{5x+7}}\)

Answer 1

Let I = \(\int\frac{dx}{\sqrt x\sqrt{5x+7}}\) = \(\int\frac{\sqrt{5x+7}}{(5x+7)\sqrt x}dx\) 

Let 5x + 7 = t²

then 5dx = 2tdt

⇒ dx = 2t/5 dt

and \(\sqrt x=\sqrt{\frac{t^2-7}5}\) = \(\frac1{\sqrt5}\sqrt{t^2-7}\) 

\(\therefore\) I = \(\int\frac{t\times2/5 t dt}{t^2\times1/\sqrt5\sqrt{t^2-7}}\) = \(\frac25\times\sqrt 5\int\frac{dt}{\sqrt{t^2-7}}\) 

 = \(\frac2{\sqrt2}log|t+\sqrt{t^2-7}|+c\) (\(\because\) \(\int\frac{dx}{\sqrt{x^2-a^2}=log|x+\sqrt{x^2-a^2}|}\))

 = \(\frac2{\sqrt5}log|\sqrt{5x+7}+\sqrt{5x}|+c\) (\(\because t^2=5x+7\))

Hence, \(\int\frac{dx}{\sqrt x\sqrt{5x+7}}=\frac2{\sqrt5}log|\sqrt{5x+7}+\sqrt{5x}|+c\)