13.4PureCore

Definite Integrals

A definite integral has limits and evaluates to a single number — no “+ c” needed. Integrate as usual, then substitute the top and bottom limits and subtract.

25 min Video by Zeeshan Zamurred Integration

Edexcel AS Level Maths: 13.4 Definite IntegralsWatch the full walkthrough before the notes below.

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What you'll be able to do

Evaluate a definite integral
Use the limits correctly (top minus bottom)
Understand why the constant c cancels
Use the square-bracket notation

The method

Integrate the function (writing it in square brackets with the limits), then compute the value at the upper limit minus the value at the lower limit.

\int_{a}^{b} f (x) d x = [F (x)]_{a}^{b} = F (b) - F (a)

Top limit value minus bottom limit value.

1Integrate:

[x^{2}]_{1}^{3}

2Substitute:

3^{2} - 1^{2} = 9 - 1

Answer

8

No constant needed

The $+ c$ would appear at both limits and cancel in the subtraction, so we leave it out for definite integrals.

Tip — Definite integral ⟶ a number, no + c. Indefinite integral ⟶ a function, with + c.

Careful with signs

Take care substituting negative limits and remember it is always $F (top) - F (bottom)$ , not the other way around.

[x^{3}]_{0}^{2} = 2^{3} - 0^{3}

Answer

8

Formula recap

\int_{a}^{b} f (x) d x = F (b) - F (a)

Top minus bottom.

[F (x)]_{a}^{b}

Square-bracket notation.

no + c

It cancels in the subtraction.

Common mistakes to avoid

Computing F(bottom) − F(top).

It is upper limit minus lower limit: F(b) − F(a).

Including + c in a definite integral.

The constant cancels; leave it out.

Key takeaways

Definite integral = F(b) − F(a), giving a number.
No + c is needed — it cancels.
Substitute the upper limit first, then subtract the lower.

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