Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 17 / a / i

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- Solution i

Solution

Write

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

and

J = \int e^{x} cos x d x

. Two applications of integration by parts, first taking the sine and then the cosine as the differentiated factor, give

I = e^{x} sin x - J + C_{1}, J = e^{x} cos x + I + C_{2} .

Thus

I + J = e^{x} sin x

and

J - I = e^{x} cos x

, in each case up to an arbitrary constant of integration. Keeping that freedom avoids identifying independently chosen antiderivatives too rigidly.

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