Why do we add + c when integrating?

Differentiating any constant gives zero, so infinitely many functions share the same derivative. The constant of integration keeps that whole family in the answer until a boundary condition fixes it.

What is the difference between definite and indefinite integration?

Indefinite integration returns a function plus a constant. Definite integration evaluates that function between two limits and returns a single number.

MathematicsA Level & IB

What is integration?

Integration is the reverse of differentiation: it recovers a function from its rate of change and measures the area under a curve.

Integration explained

An indefinite integral gives a family of functions and always needs the constant of integration,

+c

. A definite integral has limits and returns a number — usually an area, a volume of revolution or a total accumulated quantity.

Standard techniques at A Level are reverse power rule, integration by substitution, integration by parts, and the use of partial fractions for algebraic fractions.

Key formula

\displaystyle\int x^{n}\,dx = \frac{x^{n+1}}{n+1} + c \quad (n \neq -1)

Worked example

\displaystyle\int_{0}^{2} 3x^{2}\,dx = \left[x^{3}\right]_{0}^{2} = 8 - 0 = 8

, so the area under

y = 3x^{2}

between

0

and

2

8

square units.

Examiner tip

Areas below the x-axis integrate to negative values. If a question asks for total area, split the integral at every root and add the absolute values.

Integration: common questions

Why do we add + c when integrating?: Differentiating any constant gives zero, so infinitely many functions share the same derivative. The constant of integration keeps that whole family in the answer until a boundary condition fixes it.
What is the difference between definite and indefinite integration?: Indefinite integration returns a function plus a constant. Definite integration evaluates that function between two limits and returns a single number.

Related terms

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Definite integral

A definite integral evaluates an integral between two limits and returns a single numerical value, usually representing signed area.

Area under a curve

The area under a curve is the region between the graph and the x-axis, found exactly by definite integration or estimated using the trapezium rule.

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