How to integrate $x^{1/2}e^{-x}$ using integration by parts?
Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$
How to integrate $x^{1/2}e^{-x}$ using integration by parts?
Answer should be $\left(-\sqrt{x} e^{-x}+(1/2)\sqrt{\pi} \mbox{erf}(\sqrt{x})\right)+c$
I am not familiar with using e.r.f but i shall tell you the procedure. Take $u= \sqrt{x}$ and $dv = e^{-x}$. So you have $du = \frac{1}{2 \cdot \sqrt{x}} \ \rm{dx}$ and $v = -e^{-x}$. Now using the integration formula we have,
\begin{align*} \int \sqrt{x} e^{-x} \ \textrm{dx} &= \Bigl[ u \cdot v ] - \int v \cdot \ \textrm{du} \\ & = \Bigl [ -\sqrt{x} \cdot e^{-x}\Bigr] + \frac{1}{2} \int \frac{e^{-x}}{\sqrt{x}} \ \textrm{dx} \end{align*}
In the comment:
How you get erf(\sqrt(x)) from there without substitutions
If you leave the $1/2$ in the integral and do the most obvious substitution (recall your first step?), you will get something that should look like the definition of the function in @t.b.'s link.