Evaluate \[ \int x\arcsin\!\left(\frac1x\right)\,dx \quad (x>1), \] \[ \int x\arctan x\,dx, \qquad \int_0^1x\arctan x\,dx, \] and \[ \int x\sin(2x)\,dx. \]
Solution
For the first integral, integration by parts with \(u=\arcsin(1/x)\) and \(dv=x\,dx\) gives \[ \boxed{\int x\arcsin\!\left(\frac1x\right)\,dx =\frac12\left(x^2\arcsin\!\left(\frac1x\right)+\sqrt{x^2-1}\right)+C.} \]
For the second, \[ \boxed{\int x\arctan x\,dx =\frac12\left((x^2+1)\arctan x-x\right)+C.} \] Therefore \[ \boxed{\int_0^1x\arctan x\,dx=\frac\pi4-\frac12.} \]
Finally, integration by parts with \(u=x\) and \(dv=\sin(2x)dx\) gives \[ \boxed{\int x\sin(2x)\,dx =-\frac12x\cos(2x)+\frac14\sin(2x)+C.} \]