A fun practice of double integration by parts. Evaluate: \[\begin{aligned} -\int_R \frac{\partial^2}{\partial y^2}\left(\frac{t}{t^2 + y^2}\right)\cos y\,dy \end{aligned}\]

Solution:

\[\begin{aligned} -\int_R \frac{\partial^2}{\partial y^2}\left(\frac{t}{t^2 + y^2}\right)\cos y\,dy &= -\int_R \frac{\partial}{\partial y}\left(\frac{\partial}{\partial y}\left(\frac{t}{t^2 + y^2}\right)\right)\cos y\,dy \\ &= -\frac{\partial}{\partial y}\left(\frac{t}{t^2 + y^2}\right)\bigg\vert_{y=-\infty}^{\infty} - \int_R \frac{\partial}{\partial y}\left(\frac{t}{t^2 + y^2}\right) \sin y\,dy \\ &= -\int_R \frac{\partial}{\partial y}\left(\frac{t}{t^2 + y^2}\right) \sin y\,dy \\ &= -\left(\frac{t}{t^2 + y^2}\right) \sin y \bigg\vert_{y=-\infty}^{\infty} + \int_R \left(\frac{t}{t^2 + y^2}\right) \cos y\,dy \\ &= \int_R \frac{t\cos y}{t^2 + y^2} \,dy \end{aligned}\] \(\Box\)