Chapter H, Problem 50E

1. (a) If u is a complex-valued function of a real variable, its indefinite integral ${\displaystyle \int u(x)}dx$ is an antiderivative of u. Evaluate

${\displaystyle \int e^{(1+i)x}dx}$

1. (b) By considering the real and imaginary parts of the integral in part (a), evaluate the real integrals

${\displaystyle \int e^x\cos xdx}\text{and}{\displaystyle \int e^x\sin xdx}$

1. (c) Compare with the method used in Example 7.1.4.