Question

Evaluate the integral : $\int \frac{dy}{\left(y+6\right){\left(y+5\right)}^{1/2}}$ The result is

• A. $2ta{n}^{-1}\left(\sqrt{\left(y+6\right)}\right)+$ constant
• B. $2ta{n}^{-1}\left(y+6\right)+$ constant
• C. $2ta{n}^{-1}\left(\sqrt{\left(y+5\right)}\right)+$ constant
• D. $2ta{n}^{-1}\left(y+5\right)+$ constant

Answer 1

Answer: (C)

Let $I=\int \frac{dy}{\left(y+6\right)\sqrt{y+5}}$
put $y+5=t$ SO that $dy=dt$
$\therefore I=\int \frac{dt}{\left(t+1\right)\sqrt{t}}$
$=\int \frac{dt}{\left[{\left(\sqrt{t}\right)}^2+1\right]\sqrt{t}}$
Now put $\sqrt{t}=z$
$\Rightarrow \frac{1}{2\sqrt{t}}dt=dz$
$\therefore I=\int \frac{2dz}{z^2+1}$
$=2ta{n}^{-1}z+c$
$=2ta{n}^{-1}\sqrt{t}+c$
$=2ta{n}^{-1}\sqrt{y+5}+c$