Practice · Edexcel 9MA0 Pure Year 2
Integration: choose the method
Pick the method for each integral, then check the working.
Integral 1 of 8
Which method first?
What to look for
- Substitution or reverse chain ruleA function and (a multiple of) its derivative appear together, including f′(x)/f(x), which integrates to ln|f(x)|.Practise it
- Integration by partsA product of two unrelated functions, such as x with eˣ, sin x or ln x. Differentiate the part that gets simpler.Practise it
- Partial fractions, then logarithmsA fraction of polynomials whose denominator factorises, and whose top is not a multiple of the denominator's derivative.Practise it
- Use a trig identity firstPowers like sin²x or cos²x: rewrite with a double-angle identity, then integrate term by term.Practise it
Questions people ask
How do I know when to use integration by parts?
When the integrand is a product of two unrelated functions, such as x eˣ or x ln x. Choose u as the part that gets simpler when differentiated; ln x is usually u because it has no simple integral.
When should I use substitution instead?
When one part is (a multiple of) the derivative of another part, such as 2x(x² + 1)⁵. The f′(x)/f(x) pattern is the same idea and integrates to ln|f(x)|.
When do I need partial fractions?
For a fraction of polynomials whose denominator factorises and whose top is not a multiple of the bottom's derivative. Split it, then integrate each term: a linear denominator gives a logarithm.
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