Teaching Notes

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It is very helpful for students to understand that the process of integration (i.e. finding an antiderivative of an expression) is the reverse of differentiation. But I think it is also helpful for students to look back at differentiation and understand that it is basically about finding rates of change, and that integration is basically about accumulating small changes. See the page Introducing integration for discussion, ideas and examples on early days of teaching integration.

Integration basics primarily refers to relatively straightforward methods of finding indefinite integrals - mostly applying what are listed as "standard" integrals in the formula booklet. This does not include finding anti-derivatives by means of a sophisticated substitution or by the method of 'integration by parts' (HL).  However, it is certainly possible that the application of some of the standard integrals for HL could be rather 'tricky' (see questions 3 & 4 below in the set of '4 Questions'). In the formula booklet, there are five standard integrals listed for SL, and eight listed for HL (includes the 5 listed for SL) See box below showing these 'standard' integrals. Another 'straightforward' (basic) method is finding an indefinite integral by a 'simple' substitution - that is, finding the anti-derivative of a relatively uncomplicated composite function that does not require application of the formal u-substitution technique. In other words, there are anti-derivatives that students could find by using u-substitution but are simple enough to be found more efficiently by 'inspection', or by 'guess-and-check' (aka 'backwards' chain rule)

An example of an indefinite integral that a student should find by inspection is: \(\displaystyle\int {\sin \left( {3x} \right)\,{\textrm{dx}}} \)

I like to refer to the process of performing the anti-differentiation for this indefinite integral as the "backwards chain rule."

It's important to impress upon students that integration is often trickier than differentiation. And, hence, students - in my opinion - need a lot of manual practice in finding anti-derivatives of expressions, i.e. finding indefinite integrals. After introducing the basics of anti-differentiation, the first assignment I give students is a set of 15 indefinite integrals that they should answer by "inspection" or a "guess-and-check" process.

Here is a nice 'thinking' question (image left) involving integration that is suitable for both HL and SL students. Students should recognize that they can make the integration easier by first performing a trig substitution.

It is very important to emphsize that the area between a graph and the x-axis is not (necessarily) equivalent to a definite integral. Students should have this understanding illustrated and confirmed by working through problems where a curve is not always above the x-axis, as it is in the problem shown at left.

4 questions - ‘accessible’ to ‘discriminating’

download: 4_Qs_integration_basics_1_with_answers

 ♦ teaching materials

AA_SL_5.10_int_by_inspection_v1 
Set of 15 indefinite integrals to be performed either by 'inspection' or a simple 'guess-and-check' process. Answers included

AA_SL_5.10_integration_practice1_v1 
Set of 20 exercises - 10 indefinite integrals & 10 definite integrals - that cover applying rules for ‘standard’ integrals and integration by simple substitution. Answers included

AA_SL_5.10_integration_exam-like_Qs_v1 
Three reasonably challenging exam-like (Paper 2) questions involving integration; including area under a curve and volume of solid of revolution (region rotated about x-axis). GDC is allowed on all 3 questions. Worked solutions / notes available below.

AA_SL_5.10_integration_exam-like_Qs_SOL_KEY_v1 
Detailed worked solutions / notes for the 3 integration exam-like Qs from above. Good use of GDC throughout.