Indefinite & definite integrals
An indefinite integral is a family of antiderivatives, hence the +C; a definite integral is a number, computed by the fundamental theorem of calculus as F(b) − F(a). The theorem is what links the two ideas — area under a curve and reversing differentiation — and it is why the constant cancels and can be ignored in definite calculations.
✓ Unlimited questions · marked criterion by criterion · no card needed
Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Reverse the power rule∫xⁿ dx = x^(n+1)/(n+1) + C for n ≠ −1. The exception n = −1 gives ln|x| + C.
- Always write +C for indefinite integralsIt is a marked item, and omitting it is treated as a genuine error.
- Apply the fundamental theorem for definite integralsFind any antiderivative F and compute F(b) − F(a). The constant cancels.
- Interpret the signA definite integral is signed area: regions below the axis contribute negatively. For total area, split at the roots.
Worked example
Evaluate ∫₀² (3x² − 2x + 1) dx.
- Antiderivative term by term: x³ − x² + x.
- Evaluate at the upper limit: 8 − 4 + 2 = 6.
- Evaluate at the lower limit: 0 − 0 + 0 = 0.
- Subtract: 6 − 0.
Answer. 6.
Where marks get dropped
These are the specific errors that cost credit on indefinite & definite integrals questions — QED's rubric penalises each of them separately.
- Omitting +C from an indefinite integral.
- Using x^(n+1)/(n+1) when n = −1, which divides by zero. ∫ x⁻¹ dx = ln|x| + C.
- Reporting a negative definite integral as an area. Signed area is not the same as total area — split at the zeros first.
Practise this until it is automatic
Unlimited fresh questions
QED generates new indefinite & definite integrals problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.
Marked like an examiner
Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.
Answer in real notation
A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.
Saved to your library
Every question you generate is kept and re-takeable as a timed exam, and your Calculus mastery is tracked so you know when this is exam-ready.
Indefinite & definite integrals — frequently asked questions
Why does the constant not matter for definite integrals?
Because (F(b)+C) − (F(a)+C) = F(b) − F(a). Any antiderivative gives the same answer.
What are the two parts of the fundamental theorem?
Part one says d/dx ∫ₐˣ f(t)dt = f(x); part two says ∫ₐᵇ f = F(b) − F(a). Together they make differentiation and integration inverse operations.
How do I find total area rather than signed area?
Find where f crosses the axis, integrate over each piece, and add the absolute values.
The rest of Calculus
Limits, derivatives, integrals and their applications. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Limits & continuity
- 2Product, quotient & chain rules
- 3Tangent lines & linear approximation
- 4Extrema & optimisation
- 5Indefinite & definite integrals
- 6Substitution & integration by parts
- 7Series & convergence basics
- 8Implicit differentiation & related rates
- 9Mean value & intermediate value theorems
- 10Improper integrals & convergence
- 11Taylor & Maclaurin polynomials
- 12L’Hôpital’s rule & indeterminate forms
- 13Curve sketching: asymptotes, concavity & inflection
Ready to make indefinite & definite integrals exam-proof?
Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.
Start practising free →