A block of mass \(m\) is initially at rest on a frictionless horizontal \(xy\)-plane. During the time interval \(0 \le t \le T\), a horizontal force \(\vec{F}(t)\) is applied to the block with a time-dependent magnitude \(F(t) = F_{\text{max}} \sin\left(\dfrac{\pi t}{T}\right)\), where \(F_{\text{max}}\) and \(T\) are positive constants. The direction of the force rotates in the plane such that the angle it makes with the \(+x\)-axis is given by \(\theta(t) = \dfrac{\pi t}{2T}\). Which of the following expressions correctly represents the integral setup for the \(x\)-component of the impulse, \(J_x\), delivered to the block over the interval \(0 \le t \le T\)?