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Integration — standard integrals and substitution by inspection (IB Math AI HL)
Updated October 2026
Quick answer: HL AI integrates xⁿ (n ≠ −1), 1/x, eˣ, sin x and cos x, and integrates by inspection when the integrand is a function of g(x) multiplied by a multiple of g′(x). Definite integrals give areas, and the GDC evaluates any definite integral you can set up.
HL AI integrates xⁿ, sin x, cos x, 1/x and eˣ, and uses substitution (by inspection) for composites such as ∫2x(x² + 1)³ dx. Integration by parts is AA HL only, not HL AI.
Practise Integration — standard integrals and substitution by inspection with AI marking
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∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, and ∫(1/x) dx = ln|x| + C.
∫eˣ dx = eˣ + C, ∫sin x dx = −cos x + C and ∫cos x dx = sin x + C. Angles are in radians.
By inspection: if the integrand is k × g′(x) × f(g(x)), guess the answer from f, differentiate it to check, and adjust the constant. For example, ∫cos(3x) dx = (1/3) sin(3x) + C.
A fraction whose numerator is a multiple of the derivative of the denominator integrates to a logarithm: ∫g′(x)/g(x) dx = ln|g(x)| + C.
Add + C to every indefinite integral; a definite integral has no + C.
Area between a curve and the y-axis: integrate x with respect to y, between the y-limits.
Worked example
Find ∫x/(x² + 4) dx, and hence find the exact value of ∫₀² x/(x² + 4) dx.
The derivative of x² + 4 is 2x, and the numerator is x = ½ × 2x.
So ∫x/(x² + 4) dx = ½ ln(x² + 4) + C. (Check: d/dx of ½ ln(x² + 4) is ½ × 2x/(x² + 4) = x/(x² + 4).)
Definite integral: ½ ln(2² + 4) − ½ ln(0² + 4) = ½ ln 8 − ½ ln 4.
= ½ ln(8/4) = ½ ln 2 ≈ 0.347.
Answer: ½ ln(x² + 4) + C; the exact value is ½ ln 2 ≈ 0.347.
It means recognising the answer instead of using a formal substitution: if the integrand is a multiple of g′(x) times a function of g(x), you can write down the integral of that function of g(x), then differentiate your answer to check the constant.
Where can I practise integration questions?
On the Advanced Integration and Kinematics questions page: 31 exam-style HL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for integration?
Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working. The Numerical derivative and integral and Trapezium rule guides give the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.