 Solution

ID: past-exam-of-the-integration-course-of-the-university-of-oxford/2021/2/b/ii/solution

Again

∣ e^{- a x} sin (b x) ∣ \leq e^{- a x}

proves Lebesgue integrability. Taking the imaginary part of the absolutely convergent complex integral in part i gives

\int_{0}^{\infty} e^{- a x} sin (b x) d x = \frac{b}{a ^{2} + b ^{2}} .

 New to topics? Read the docs here!