Topic 5: Integration Techniques

3 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does splitting a rational function into simpler fractions make it integrable?

Decompose rational functions into partial fractions and use the decomposition to integrate them.

Decomposing a proper rational function into partial fractions over linear factors, finding the unknown constants, and integrating the pieces to logarithms.

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How does reversing the product rule let us integrate products such as x times e to the x?

Evaluate integrals using integration by parts, including choosing u and dv and applying the method more than once where needed.

Integration by parts as the reverse of the product rule, choosing u and dv with the LIATE guide, repeated application, and the standard trick for integrating the natural logarithm.

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How does reversing the chain rule let us integrate composite functions?

Evaluate integrals using the method of substitution, including changing the limits for definite integrals.

Using substitution to reverse the chain rule, choosing u and replacing dx with du, handling definite integrals by changing the limits, and recognising when substitution applies.

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