Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 4 / a

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- Solution a

Solution

Let

I = \int e^{x} sin x d x

. Two applications of integration by parts give

I = e^{x} sin x - \int e^{x} cos x d x, \int e^{x} cos x d x = e^{x} cos x + I,

where additive constants can be absorbed at the end. Solving for

I

yields

\int e^{x} sin x d x = \frac{e ^{x}}{2} (sin x - cos x) + C .

Differentiating the answer gives

e^{x} sin x

.

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