SCalc97.1.AE.004.EXAMPLE 4 Evaluate e5xsin(x)dx.SOLUTION Neither e5x nor sin(x) becomes simpler when differentiated, but we try choosing u=e5x and dy=sin(x)dx amyway. Then dt=,dx and y=,, so integration by parts gives(1)e5xsin(x)dx=5e5xcos(x)dx.The integral that we have obtained, e5xcos(x)dx, is no simpler than the original one, but at least it is no more difficult. Having had success in the preceding example integrating by parts twice, we persevere and integrate by parts again. This time we use u=e5x and dv=dx. Then du=5e5xdx,v=sin(x), and(7)e5xcos(x)dx=-5e5xsin(x)dx.At first glance, it appears as if we have accomplished nothing because we have arrived at e5xsin(x)dx, which is where we started. However, if we put the expression for cos(x)e5xdx from Equation (2) into Fquation (11) we get