Question

If $\int \sqrt{\frac{x-5}{x-7}}dx=A\sqrt{x^2-12x+35}+\log |x-6+\sqrt{x^2-12x+35}|+C$ then $A=$

• A. $-1$
• B. $\frac{1}{2}$
• C. $-\frac{1}{2}$
• D. $1$

Answer 1

The correct option is D $1$

Given : $\int \sqrt{\frac{x-5}{x-7}}dx=A\sqrt{x^2-12x+35}+\log |x-6+\sqrt{x^2-12x+35}|+C$ ...... $\left(i\right)$

$\int \sqrt{\frac{x-5}{x-7}}dx$

$=\int \frac{x-5}{\sqrt{x^2-12x+35}}dx$ ...... [Using rationalisation method]

$=\frac{1}{2}\int \frac{2x-10}{\sqrt{x^2-12x+35}}dx$
$=\frac{1}{2}\int \frac{2x-12+2}{\sqrt{x^2-12x+35}}dx$

$=\frac{1}{2}\int \frac{2x-12}{\sqrt{x^2-12x+35}}dx+\frac{dx}{\sqrt{x^2-12x+36-1}}$

$=\frac{1}{2}2\sqrt{x^2-12x+35}+\int \frac{dx}{\sqrt{(x-6{)}^2-1}}+c_1$

$=\sqrt{x^2-12x+35}+\log |x-6+\sqrt{x^2-12x+35}|+c$

Comparing above equation with $\left(i\right)$, we get

$A=1$