Just want to share my work here. 1. ∫ e^x sin(ax) dx Let I = ∫ e^x sin(ax) dx =^(IBP) e^x sin(ax) − a∫ e^x cos(ax) dx =^(IBP) e^x sin(ax) − a[e^x cos(ax) + a∫ e^x sin(ax) dx] = e^x sin(ax) − ae^x cos(ax) − a^2 I I = e^x sin(ax) − ae^x cos(ax) − a^2 I (a^2 + 1)I = e^x sin(ax) − ae^x cos(ax) I = ((e^x sin(ax))/(a^2 + 1)) − ((ae^x cos(ax))/(a^2 + 1)) + C determinant (((∫ e^x sin(ax) dx = ((e^x sin(ax))/(a^2 + 1)) − ((ae^x cos(ax))/(a^2 + 1)) + C))) 2. ∫ e^x cos(ax) dx Let I = ∫ e^x cos(ax) dx =^(IBP) e^x cos(ax) + a∫ e^x sin(ax) dx =^(IBP) e^x cos(ax) + a[e^x sin(ax) − a∫ e^x cos(ax) dx] = e^x cos(ax) + ae^x sin(ax) − a^2 I I = e^x cos(ax) + ae^x sin(ax) − a^2 I (a^2 + 1)I = e^x cos(ax) + ae^x sin(ax) I = ((e^x cos(ax))/(a^2 + 1)) + ((ae^x sin(ax))/(a^2 + 1)) + C determinant (((∫ e^x cos(ax) dx = ((e^x cos(ax))/(a^2 + 1)) + ((ae^x sin(ax))/(a^2 + 1)) + C))) Note: C is an arbitrary constant Open for corrections.