Integration by parts, substitution and differential equations
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 1 worked example, 4 practice questions, 2 real past-paper questions with the answers explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- The tools beyond the basics
- Integration by parts
Read the transcript of the teaching part
Last time
Before we start: last time, in Implicit and parametric differentiation, quotient rule, and stationary points with integration and iteration, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
The tools beyond the basics
Some integrals look impossible: x times e to the 3x, or one over x squared plus a bracket. Each has a standard attack. Parts undoes the product rule, substitution undoes the chain rule, and algebra, such as division or partial fractions, tidies a fraction before you integrate it.
Integration by parts
The formula: the integral of u dv by dx equals u v, minus the integral of v du by dx. Choose u to be the part that gets simpler when differentiated, since that makes the new integral easier, such as x or x squared or a logarithm. Let v dash be the part you can integrate, such as e to the x or sine x. If you must integrate a logarithm or an inverse tangent on its own, treat it as the log times one.
How often it comes up
How often Cambridge has asked each part, counted by ExamYard from every past paper in its question bank: Integration by parts · Integration by substitution · Integration using trig identities · Partial fractions and integration · Separating the variables.
The whole chapter: Differential Equations, Integration.
Chapter 3.5: Integration
- Integration by parts, substitution and differential equations 7 min
Also for A Level Maths 9709 Paper 3
Which topics come up most in A Level Maths 9709 Paper 3 past papers, counted from every paper since 2004.