Integration techniques extend the standard integrals from earlier in the H2 syllabus to integrands that are not immediately recognisable as a standard result. You first manipulate the expression by substitution, by parts, by splitting into partial fractions or by using trigonometric identities. Then a standard integral can be applied.
Before attempting any technique, always check whether the integrand is already a standard form. Students most often fail to spot the f′(x) ÷ f(x) rule. If the numerator is (a multiple of) the derivative of the denominator, the integral is a natural log and no substitution is required.
What these notes cover
- •Standard integrals (recap)
- •The f′(x) ÷ f(x) rule
- •Integration by substitution
- •Trigonometric substitution for definite integrals
- •Integration by parts and LIATE
- •Applying integration by parts twice
- •Rational functions and long division
- •Partial fractions: linear, repeated linear and quadratic factors
- •Integrating with double-angle identities
- •Product-to-sum identities
- •Common mistakes
- •Summary
Key definitions
- Integration by substitution
- Replacing an awkward part of the integrand with u = g(x). Use it when the integrand contains an inner function whose derivative (up to a constant multiple) also appears, or when a surd or composite expression is blocking a standard form.
- Integration by parts
- A technique for integrating a product of two functions of different “families” that is not simply a chain-rule derivative.
- LIATE
- A priority order for choosing u (the factor to differentiate) in integration by parts: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential. The function higher on the list should usually be u, since it becomes simpler when differentiated.
- f′(x) ÷ f(x) rule
- If the numerator is (a multiple of) the derivative of the denominator, the integral is a natural log, ln|f(x)| + C, with no substitution required.
- Partial fractions
- Splitting a proper fraction P(x) ÷ Q(x) into simpler fractions. Each linear or repeated linear factor of the denominator takes a constant numerator, and each irreducible quadratic factor takes a linear numerator.
Key formulas
Exponential of a linear function
Standard result; a and b are constants and C is the constant of integration.
The f′(x) ÷ f(x) rule
Applies whenever the numerator is (a multiple of) the derivative of the denominator.
Integration by parts
u is the factor you differentiate (chosen by LIATE order). dv/dx is the factor you integrate to get v.
Repeated linear factor term
The second partial fraction from a repeated factor (x − a)² integrates by the power rule, not the log rule.
Double-angle rewrites
sin²x = ½(1 − cos 2x), cos²x = ½(1 + cos 2x)
Use these to rewrite even powers of sin or cos before integrating. Both functions must share the same argument.
Product-to-sum identity
2 sin A cos B = sin(A + B) + sin(A − B)
Use for products of sines and cosines with different arguments, e.g. sin 3x cos x with A = 3x and B = x.
Worked example
Find ∫ (2x² + 3x − 1) ÷ ((x − 1)(x² + 1)) dx.
- 1Let (2x² + 3x − 1) ÷ ((x − 1)(x² + 1)) = A ÷ (x − 1) + (Bx + C) ÷ (x² + 1).
- 2Multiplying through gives 2x² + 3x − 1 = A(x² + 1) + (Bx + C)(x − 1).
- 3Setting x = 1 gives 4 = 2A, so A = 2.
- 4Comparing x² coefficients gives 2 = A + B, so B = 0.
- 5Comparing constants gives −1 = A − C, so C = 3.
- 6The integral becomes ∫ (2 ÷ (x − 1) + 3 ÷ (x² + 1)) dx = 2 ln|x − 1| + 3 arctan(x) + C.
Watch out: Do not mismatch the numerator form to the factor type. A linear numerator (Bx + C) goes over an irreducible quadratic factor, and a constant goes over each linear (or repeated linear) factor.
Common mistakes
Integrating sin²x with the power rule
Wrong: ∫ sin²x dx = sin³x ÷ 3 + C
Right: Rewrite with a double-angle identity first: ∫ sin²x dx = ∫ ½(1 − cos 2x) dx = ½x − ¼ sin 2x + C.
Choosing u the wrong way round in integration by parts
Wrong: Choosing u = eˣ over u = x. This makes the resulting integral more complicated.
Right: Check LIATE order. Algebraic comes before Exponential, so take u = x and dv/dx = eˣ.
Missing the second term for a repeated linear factor
Wrong: Using only A ÷ (x − a) for a factor (x − a)², or integrating B ÷ (x − a)² to a ln term.
Right: Use A ÷ (x − a) + B ÷ (x − a)². The second term integrates by the power rule to −B ÷ (x − a).
Questions students ask
How do I know which integration technique to use in H2 Math?
First check whether the integrand is already a standard form, including the f′(x) ÷ f(x) rule. Use substitution when an inner function and its derivative both appear, and integration by parts for a product of two functions of different families. Use partial fractions for rational functions, and trigonometric identities for even powers or products of sin and cos.
How do you choose u in integration by parts?
Use the LIATE priority order (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) and take the function higher on the list as u, since it becomes simpler when differentiated. Some integrals, such as ∫ eˣ sin x dx, need integration by parts twice. The original integral then reappears on the right, so collect like terms and solve for it algebraically.
How do you integrate a fraction with x² + 1 in the denominator?
If deg(P) ≥ deg(Q), divide first. Then use partial fractions, with a linear numerator (Bx + C) over the irreducible quadratic factor and a constant over each linear factor. The x-part of the numerator typically gives a ln term and the constant part gives an arctan term.
Do I need to change the limits when using substitution on a definite integral?
Yes. Change the limits to u-limits, and you can then skip the back-substitution. With a trigonometric substitution, such as x = a sin θ for √(a² − x²) or x = a tan θ for a² + x², change the limits to θ and simplify the surd using the corresponding Pythagorean identity.
In the full PDF
- •A step-by-step worked example by substitution: ∫ x√(3x − 2) dx
- •A worked example by parts: ∫ (x + 1)e³ˣ dx
- •A worked example using product-to-sum: ∫ sin 3x cos x dx
- •All three product-to-sum identities and the tan²x = sec²x − 1 rewrite
- •The remaining common mistakes, including double-angle misuse and half-converted substitutions
- •An end-of-topic summary of every technique