Integrating using Complex numbers

The traditional method to integrate these functions is through integration by parts. However by using complex numbers one can integrate them even faster. $\int e^{ax}(\cos(bx)dx+i\int e^{ax}\sin(bx))dx$ $\int e^{ax}(\cos(bx)+i\sin(bx))dx$ $\Rightarrow \int e^{ax}e^{ibx}dx$ $\Rightarrow \int e^{(a+ib)x}dx$ $\Rightarrow \frac{e^{(a+ib)x}}{a+ib}+C$ $\Rightarrow \frac{e^{ax}e^{ibx}(a-ib)}{a^2+b^2}+C$ $\Rightarrow \frac{e^{ax}(\cos(bx)+i\sin(bx))(a-ib)}{a^2+b^2}+C$ $\Rightarrow \frac{e^{ax}(a\cos(bx)+b\sin(bx)+i(a\sin(bx)-b\cos(bx)))}{a^2+b^2}+C$ $\Rightarrow \frac{e^{ax}(a\cos(bx)+b\sin(bx))}{a^2+b^2}+i\frac{e^{ax}(a\sin(bx)-b\cos(bx))}{a^2+b^2}+C$ Thus $\int e^{ax}\cos(bx)dx=\frac{e^{ax}(a\cos(bx)+b\sin(bx))}{a^2+b^2}+C$ and $\int e^{ax}\sin(bx)dx=\frac{e^{ax}(a\sin(bx)-b\cos(bx))}{a^2+b^2}+C$