25 August 2026
A-Level Maths Integration: How to Choose the Right Method
A decision-tree approach to A-Level Maths integration, covering standard integrals, substitution, integration by parts, partial fractions, trig identities and differential equations.
Integration causes more confusion at A-Level than differentiation, not because the techniques themselves are hard, but because choosing which technique to use is a separate skill that isn't always taught explicitly. Differentiation is largely mechanical — you apply a fixed set of rules. Integration requires you to recognise the shape of an expression and match it to a method. This guide gives you a practical decision process to follow, in order, whenever you meet an integral you don't immediately recognise.
If you also want to tighten up related exam technique — for example remembering the constant of integration or handling definite integral limits correctly — it's worth reading our article on common mistakes in A-Level Maths exams alongside this one, since integration errors feature heavily in examiners' reports.
Step 1: Is it a standard integral?
Before reaching for any technique, check whether the expression matches a standard result you should recognise instantly, or one listed in the formula booklet (see our guide to what's given and what you must memorise). This includes:
- Powers of x: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c
- Exponentials: ∫eᵏˣ dx = (1/k)eᵏˣ + c
- ∫1/x dx = ln|x| + c
- ∫sin x dx, ∫cos x dx, ∫sec²x dx
- ∫1/(1+x²) dx and similar forms leading to inverse trig functions
If the integral is a straightforward standard form, or a linear transformation of one (such as ∫sin(3x + 2) dx), integrate directly using the reverse chain rule and don't overcomplicate it with substitution.
Step 2: Does the integrand look like the result of a chain rule differentiation?
If you can see a function and what looks like its derivative multiplied alongside it — for example ∫2x(x² + 1)⁵ dx — this is a strong signal for integration by substitution (or recognising the reverse chain rule directly). The general pattern is:
∫f'(g(x)) · g'(x) dx = f(g(x)) + c
When to use substitution explicitly: if the "spotting" method feels unclear, set u equal to the inner function, find du/dx, and rewrite the whole integral in terms of u. This works particularly well for integrals involving composite functions, roots of linear expressions, and trigonometric functions raised to powers.
Worked example pattern: ∫x√(x+1) dx — let u = x + 1, so x = u − 1 and dx = du. Rewrite entirely in u, expand, integrate term by term, then substitute back.
Step 3: Is it a product of two different types of function?
If the integrand is a product of an algebraic function and a trigonometric, exponential or logarithmic function — such as ∫x sin x dx, ∫x eˣ dx or ∫ln x dx — this is the signal for integration by parts:
∫u (dv/dx) dx = uv − ∫v (du/dx) dx
The key skill is choosing which factor is u and which is dv/dx. A useful priority order (sometimes remembered as LIATE — Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) suggests choosing u as the function that appears earlier in that list, since it typically simplifies on differentiation. For ∫ln x dx specifically, treat ln x as u and 1 as dv/dx — this is a classic case students miss because there appears to be only one function.
Some integrals require applying integration by parts twice, particularly those involving eˣ multiplied by sin x or cos x, where the pattern eventually repeats and allows you to solve for the original integral algebraically.
Step 4: Is it a rational function (a fraction of polynomials)?
If you're integrating something like (3x + 5)/((x-1)(x+2)), this is a signal for partial fractions. The method:
- Check the degree of the numerator against the denominator — if the numerator's degree is equal to or higher, perform polynomial division first.
- Split the fraction into simpler fractions with unknown constants over each factor of the denominator.
- Multiply through and compare coefficients (or substitute convenient values of x) to find the constants.
- Integrate each simpler fraction separately — these will typically produce ln terms.
Watch for repeated linear factors (which need an extra term with the factor squared in the denominator) and irreducible quadratic factors (which need a linear numerator over them, often leading to an arctan term).
Step 5: Does it involve trigonometric powers or products?
Integrals like ∫sin²x dx, ∫cos³x dx or ∫sin x cos 3x dx cannot be integrated directly and need a trig identity rewrite first:
- For even powers of sin or cos, use the double angle formulae (cos 2x = 1 − 2sin²x or 2cos²x − 1) to rewrite in terms of cos 2x, which integrates directly.
- For odd powers, split off one factor and use sin²x + cos²x = 1 to convert the rest, then substitute.
- For products of sines and cosines with different arguments, use the appropriate sum-to-product identities from the formula booklet.
Step 6: Definite integrals and finding areas
Once you've chosen and applied the correct method, definite integrals follow the same process but you evaluate at the limits rather than adding "+ c". Common pitfalls here include:
- Forgetting to change the limits when you substitute (if you let u = g(x), the limits must be converted into u-values, not left as x-values)
- Not accounting for regions where the curve dips below the x-axis, which produces a negative "area" that needs to be treated with an absolute value if you want the actual geometric area
- Confusing the area between a curve and the x-axis with the area between two curves (the latter requires integrating the difference of the two functions)
Step 7: Differential equations
Many A-Level differential equations are solved using separation of variables: rearrange so all terms in y (including dy) are on one side and all terms in x (including dx) are on the other, then integrate both sides independently, remembering the constant of integration on at least one side. After integrating, use any given boundary condition to find the constant, then rearrange to express y in terms of x if the question asks for it explicitly.
Common contexts include rates of change problems (population growth, cooling, mixing problems) — read the question carefully to correctly set up the differential equation itself before attempting to solve it; the setup is often worth more marks than the integration.
A quick decision-tree summary
| Signal in the integrand | Method to use |
|---|---|
| Matches a standard form or reverse chain rule | Direct integration |
| Composite function with its derivative present | Substitution |
| Product of algebraic and trig/exp/log function | Integration by parts |
| Fraction of polynomials | Partial fractions, then integrate |
| Powers or products of trig functions | Trig identity rewrite, then integrate |
| Rate of change described in words | Set up and separate a differential equation |
Practising the decision process
The only way this decision process becomes automatic is repeated exposure to a wide variety of integrals under timed conditions. Work through mixed integration questions from past papers without sorting them by topic first — in the real exam you won't be told which method to use, so practising method identification, not just execution, is the real skill being tested. Our resources section and textbooks listings include material organised by topic if you need to shore up any one technique before mixing them.
If a particular technique — commonly integration by parts or partial fractions with repeated factors — consistently trips you up despite practice, working through it one-to-one with a tutor via A-Level Maths Tutoring can clear up the misunderstanding far faster than repeated independent practice on the same sticking point.
Takeaway
Integration at A-Level is a pattern-recognition skill as much as a calculation skill. Work through the decision tree above in order every time you meet an unfamiliar integral: check for a standard form, look for a substitution signal, check for a product suited to parts, check for a rational function needing partial fractions, and check for trig powers needing an identity. With enough mixed practice from past papers, this sequence becomes instinctive rather than something you have to consciously work through in the exam.
Common errors within each method
Even once you've correctly identified which method to use, execution errors are common, and knowing about them in advance helps you avoid them.
In substitution, the most frequent error is forgetting to fully replace every instance of x, including inside dx itself, which must be rewritten in terms of du. A partial substitution that leaves a mix of x and u terms in the same expression is a strong sign something has gone wrong and needs to be checked before continuing.
In integration by parts, choosing u and dv/dx the wrong way round often leads to an integral that becomes more complicated rather than simpler. If your second integral, ∫v (du/dx) dx, looks harder than the one you started with, that's a signal to swap your choice of u and dv/dx and try again.
In partial fractions, a common mistake is forgetting the extra term required for a repeated linear factor in the denominator, or forgetting that an irreducible quadratic factor needs a linear (not constant) term in its numerator. Always check your partial fraction decomposition by recombining it over a common denominator and confirming it matches the original expression before integrating.
In trigonometric integrals, students sometimes apply a double angle formula to convert cos²x correctly but then forget to also convert the differential correctly if a substitution is layered on top, or lose a factor of a half when integrating cos 2x. Writing out each rewritten term explicitly, rather than trying to simplify multiple steps at once mentally, reduces this risk considerably.
Practising integration under exam conditions
Because integration technique-selection is really a skill of pattern recognition under time pressure, the best form of practice is a mixed set of integrals presented without a topic heading, timed as you would in the real exam. Many students inadvertently avoid this by working through a textbook chapter titled "integration by parts", which removes the actual challenge of deciding which method to use, since the heading has already told you.
Instead, once you're comfortable executing each technique individually, deliberately seek out mixed exercises or full past papers where integration questions appear alongside other pure maths content, and time yourself identifying the method before you commit pen to paper. If you consistently misidentify the correct method for a particular pattern — for instance, confusing when a rational function needs partial fractions versus a direct substitution — that specific confusion is worth targeting directly with a tutor via A-Level Maths Tutoring, since it often stems from a small, fixable gap in pattern recognition rather than a lack of overall ability.
Finally, remember that integration questions frequently sit inside larger multi-part problems involving areas, volumes of revolution, or differential equations modelling real-world contexts. Reading the full question before starting, so you understand what the integral you're about to compute actually represents, often prevents wasted effort on the wrong calculation entirely.