Calculus · step 6 of 13

Substitution & integration by parts

Substitution reverses the chain rule: spot an inner function whose derivative is a factor, set u equal to it, and the integral simplifies. Integration by parts reverses the product rule: ∫u dv = uv − ∫v du, and the LIATE order (logarithmic, inverse trig, algebraic, trigonometric, exponential) tells you which factor to call u.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Try substitution firstLook for a composite whose inner derivative appears as a factor, up to a constant. Set u to the inner function.
Change the limits tooFor definite integrals, convert the limits to u-values rather than substituting back — it is quicker and less error-prone.
Use LIATE for by-partsWhichever type comes first in LIATE becomes u; the rest is dv. This choice makes ∫v du simpler than what you started with.
Repeat or loop if necessary∫eˣ sin x dx returns to itself after two applications — solve the resulting equation for the integral.

Worked example

Evaluate ∫ x·eˣ dx.

LIATE: algebraic before exponential, so u = x and dv = eˣ dx.
Then du = dx and v = eˣ.
Apply ∫u dv = uv − ∫v du: x·eˣ − ∫eˣ dx.
= x·eˣ − eˣ + C.

Answer. eˣ(x − 1) + C. Differentiating back gives eˣ(x−1) + eˣ = x·eˣ ✓.

Where marks get dropped

These are the specific errors that cost credit on substitution & integration by parts questions — QED's rubric penalises each of them separately.

Substituting for u but forgetting to convert dx into du. Every part of the integral must be in terms of u.
Changing variables in a definite integral without changing the limits, then evaluating u-limits as if they were x-limits.
Choosing u and dv so that ∫v du is harder than the original. If that happens, swap them.

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Substitution & integration by parts — frequently asked questions

How do I choose a substitution?

Look for an inner function whose derivative appears as a factor. For ∫2x·e^(x²) dx take u = x², since du = 2x dx is already present.

What does LIATE stand for?

Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential — the priority order for choosing u. It is a heuristic, not a theorem, but it works nearly always.

What if by-parts loops back to the original?

Solve algebraically. For I = ∫eˣ sin x dx, two applications give I = eˣ(sin x − cos x) − I, so I = eˣ(sin x − cos x)/2 + C.

The rest of Calculus

Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.

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