$\int e^{a x} \cos b x d x$ is equal to

real part of $\int e^{(a+b i) x} d x$ imaginary part of $\int e^{(a+b i) x} d x$ neither real nor Imaginary part of $\int e^{(a+b i) x} d x$ none of these

real part of $\int e^{(a+b i) x} d x$

Let $I=\int e^{a x} \cos b x d x$ = real part of $\int e^{a x} e^{i b x} d x$ = real part of $\int e^{(a+i b) x} d x$ Hence (1) is the correct answer.