integral of (x^(1/2))e^(x^(1/2))
upper limit = 4
lower limit = 0
how do I do this?
I have found that u-substitution is the most useful method of indefinite integration
$\displaystyle \int\sqrt{x}e^{\sqrt{x}}~dx$
Let $\displaystyle \sqrt{x}=z\implies x=z^2$ so $\displaystyle dx=2z~dz$ so our integral becomes
$\displaystyle \int\sqrt{x}e^{\sqrt{x}}~dx\stackrel{\sqrt{x}=z}{\ longmapsto}2\int ze^{z}\cdot z~dz$
which can be done by parts.
$\displaystyle \begin{aligned}2\int_0^2 x^2\cdot e^x~dx&=2x^2\cdot e^x\bigg|_{x=0}^{x=2}-4\int_0^2 x\cdot e^x~dx\\
&=8e^2-\left(4x\cdot e^x\bigg|_{x=0}^{x=2}-4\int_0^2 e^x~dx\right)\\
&=8e^2-8e^2+4e^x\bigg|_{x=0}^{x=2}\\
&=4e^2-4\end{aligned}$
Im sure you can fill in the missing steps