IB Maths AI HL Ahl 5 11 Hl Further Integration Questions

Practise integrating standard functions by inspection or substitution, using definite integrals for area or change, and checking signs, limits and units.

Exam points

Integrate x^n, sin x, cos x, sec²x and e^x by inspection or substitution.
Use definite integrals and bounds to calculate accumulated change or area, handling negative integrals correctly.
Check the antiderivative by differentiation and interpret constants, limits and units.

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Study AHL 5.11 (HL)—Further integration conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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Find the indefinite integral $\int x e^{-x^{2}} d x$ .

A valid/coherent attempt to use substitution:

u=-x^2,\quad \frac{du}{dx}=-2x

The integral becomes

-\frac12\int e^u\,du=-\frac12e^{-x^2}+c

Accept integration by inspection using $\frac{d}{dx}(e^{-x^2})=-2xe^{-x^2}$ .
Note: Award M1A1M1A0 for $-\frac12e^{-x^2}$ with c omitted.

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IB Maths AI HL Ahl 5 11 Hl Further Integration Questions question 1