Use integration by parts with u = x to integrate x cos 2x…

Step 1. Parts. $u = x$, $dv = \cos 2x\,dx$, so $du = dx$ and $v = \dfrac{\sin 2x}{2}$. Then $\displaystyle\int x\cos 2x\,dx = \dfrac{x\sin 2x}{2} - \int \dfrac{\sin 2x}{2}\,dx = \dfrac{x\sin 2x}{2} + \dfrac{\cos 2x}{4}$.

Step 2. The constant. $F(0) = 0 + \dfrac{1}{4} + C = 0$, so $C = -0.25$.

Step 3. Evaluate. At $x = \dfrac{\pi}{4}$: $\sin\dfrac{\pi}{2} = 1$ and $\cos\dfrac{\pi}{2} = 0$. So $F = \dfrac{\pi}{8} + 0 - 0.25 = 0.1427$.