Answer
$$\int_0^{\sqrt \pi}x\cos(x^2)dx=0$$
Work Step by Step
To evaluate the integral $$\int_0^{\sqrt \pi}x\cos(x^2)dx$$ we will use substitution $x^2=t$ which gives us $2xdx=dt\Rightarrow xdx=\frac{dt}{2}$ and the integration bounds would be: for $x=0$ we have $t=0$ and for $x=\sqrt\pi$ we have $t=\pi$. Putting this into the integrate we get:
$$\int_0^{\sqrt \pi}x\cos(x^2)dx=\int_0^\pi\cos t\frac{dt}{2}=\frac{1}{2}\left.\sin t\right|_0^\pi=\frac{1}{2}(\sin \pi -\sin0)=\frac{1}{2}(0-0)=0$$