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Past exam of the integration course of the University of Oxford / 2021 / 2 / b / ii / Solution 2026-10-25

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}} .

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