Let $\mathrm{P}=\int {\mathrm{e}}^{\mathrm{ax}}\cos \mathrm{bxdx},\mathrm{Q}=\int {\mathrm{e}}^{\mathrm{ax}}\sin \mathrm{bxdx}\text{. If }\mathrm{P}+\mathrm{iQ}=\mathrm{Z}+\mathrm{C}$where Z is a complex function and C is constant then |Z|

$$\begin{gathered} p=\int e^{ax}\cos bxdx \\ =\frac{e^{ax}}{a^2+b^2}\left[a\cos bx+b\sin bx\right] \\ Q=\int e^{ax}\sin bx\cdot dx \\ =\frac{e^{ax}}{a^2+b^2}\left[a\sin bx-b\cos bx\right] \\ \left|Z\right|=\sqrt{P^2+Q^2}=\frac{e^{ax}}{a^2+b^2}\sqrt{{\left[a\cos bx+b\sin bx\right]}^2+{\left[a\sin bx-b\cos bx\right]}^2} \\ =\frac{e^{ax}}{a^2+b^2}\sqrt{a^2+b^2}=\frac{e^{ax}}{\sqrt{a^2+b^2}}=\frac{{\mathrm{e}}^{\mathrm{ax}}}{|\mathrm{a}+\mathrm{ib}|} \end{gathered}$$