Expression's integral domain

Question

Given the next expression:

$$f(x)=\int_{0}^{x^2}{e^{-2\sqrt{t}}}dt$$

How can i calculate its domain? I figured that it was related to its derivative, but not sure how to get there.

Help? Thanks

Answer 1

To find the domain of $f(x)$, we will begin by evaluating the integral. $$\begin{aligned} f(x)&=\int_0^{x^2}e^{-2\sqrt t}dt \\ &=\int_0^x2ue^{-2u}du;t=u^2 \\ &=\left(ue^{-2u}+\frac 12e^{-2u}\right)_0^x \\ &=\left(xe^{-2x}+\frac 12e^{-2x}\right)-\left(\frac 12e^0\right) \\ &=xe^{-2x}+\frac 12e^{-2x}-\frac 12. \end{aligned}$$ This is clearly defined over all of $\mathbb R$.