Calculate $\int_{\pi^2}^{4\pi^2} \frac{\cos(\sqrt{x})}{\sqrt{x}} \,dx$ I am trying to calculate $$\int_{\pi^2}^{4\pi^2} \frac{\cos(\sqrt{x})}{\sqrt{x}} \,dx.$$
I calculated the integral and got $2\sin(\sqrt{x})$ as a result, but for $x=\pi^2$ and $x=4\pi^2$ we get that $2\sin(\sqrt{\pi^2})=0$ and $2\sin(\sqrt{4\pi^2})=0$ So the Riemann integral will be $0-0=0$ which is not true, as you can see from ploting $2\sin(\sqrt{x})$.
Any help will be much appreciated!
 A: 
$$\int_{\pi^2}^{4\pi^2} \frac{\cos(\sqrt{x})}{\sqrt{x}} \,dx.$$

Letting $u=\sqrt{x}$, we have $du=\large\frac{1}{2\sqrt{x}}\,dx\;$ or $\;2\,du = \large\frac{dx}{\sqrt{x}},\;$ so the integral becomes
$$ \int_\pi^{2\pi} 2\cos u \,du\;=\;2\sin u\Big|_\pi^{2\pi} = 2\sin (2\pi)-2\sin(\pi)\,=\,0 - 0 = 0 $$
I simply changed the bounds of integration, so there's no need to "back-substitute". So it seems, as you proceeded in your evaluation of the definite integral, that your answer is indeed correct.
See, e.g. Wolfram|Alpha's computation:
$\quad\quad\quad\quad$

Visual representation of the integral:
 
*If we are looking to calculate the area between the x-axis and the curve, then we need to split the integral to compute the area below the x-axis, and the area above the x-axis: for $u$, the dividing point will be $\large\frac{3\pi}{2}$ (for $x$: $\large\frac{9\pi}{4}).$
$$\Big|\int_\pi^{\large\frac{3\pi}{2}} 2\cos u \,du\;\Big|\;\;+ \;\;\Big|\int_{\large\frac{3\pi}{2}}^{2\pi} 2\cos u \,du\;\Big|\;\; = \;\;\Big|2\sin u|_\pi^{3/2\pi}\Big|\;+\; \Big|2\sin u|_{3/2\pi}^{2\pi}\Big|\; =\; 2 + 2 = 4$$  
A: You are right. Here is a slightly different approach, if you want.
Using the substitution $u=\sqrt{x}$, we have $du=\frac{1}{2\sqrt{x}}dx$ so the integral becomes
$$
\int_\pi^{2\pi} 2\cos u du=2\sin (2\pi)-2\sin(\pi)=0 . 
$$
A: Although @julien got the right fact about the definite integral, we should find a positive value for that. I mean we should consider the following integrals as well to find a real area: $$
\Big|\int_\pi^{3\pi/2}\Big|\;\;+\;\;\Big|\int_{3\pi/2}^{2\pi}\Big|  
$$
