(Solved): 3. By Euler's formula \( \int e^{x+i x} \mathrm{~d} x=\int e^{x}(\cos x+i \sin x) \mathrm{d} x \). ...

3. By Euler's formula \( \int e^{x+i x} \mathrm{~d} x=\int e^{x}(\cos x+i \sin x) \mathrm{d} x \). Integrate the left-hand side and then (by hand) isolate the real and imaginary parts to find expressions for \( \int e^{x} \cos x \mathrm{~d} x \) and \( \int e^{x} \sin x \mathrm{~d} x \) (and so bypass the integration-by-parts used in M1001).