Answer
$$\int\sqrt x e^{\sqrt x}dx=2e^{\sqrt x}(x-2\sqrt x+2)+C$$
Work Step by Step
$$A=\int\sqrt x e^{\sqrt x}dx$$
We set $a=\sqrt x$, which means $$da=\frac{1}{2\sqrt x}dx=\frac{1}{2a}dx$$ $$dx=2ada$$
Therefore, $$A=\int a e^a(2ada)=2\int a^2e^ada$$
Next, we do integration by parts.
Take $u=a^2$ and $dv=e^ada$
We would have $du=2ada$ and $v=e^a$
Applying the formula $\int udv=uv-\int vdu$, we have
$$A=2(a^2e^a-2\int ae^ada)$$ $$A=2a^2e^a-4\int ae^ada$$
Take $u=a$ and $dv=e^ada$
We would have $du=da$ and $v=e^a$
Applying the formula $\int udv=uv-\int vdu$, we have
$$A=2a^2e^a-4(ae^a-\int e^ada)$$ $$A=2a^2e^a-4(ae^{a}-e^a)+C$$ $$A=2a^2e^a-4ae^{a}+4e^a+C$$ $$A=2e^a(a^2-2a+2)+C$$ $$A=2e^{\sqrt x}(x-2\sqrt x+2)+C$$
