Practicing Success
The value of $\int \frac{\sqrt{1+x}}{x} d x$, is |
$2 \sqrt{1+x}+\log \left|\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}\right|+C$ $2 \sqrt{1+x}+C$ $\log _e\left|\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}\right|+C$ $\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}+C$ |
$2 \sqrt{1+x}+\log \left|\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}\right|+C$ |
Let $I =\int \frac{\sqrt{1+x}}{x} d x$ $\Rightarrow I =\int \frac{\sqrt{t^2}}{t^2-1} 2 t d t$, where $1+x=t^2$ $\Rightarrow I =2 \int \frac{t^2-1+1}{t^2-1} d t=2 \int\left(1+\frac{1}{t^2-1}\right) d t$ $\Rightarrow I=2 t+\log \left|\frac{t-1}{t+1}\right|+C=2 \sqrt{1+x}+\log \left|\frac{\sqrt{1+x}-1}{\sqrt{1+x}+1}\right|+C$ |