Definite Integrals

Evaluating integrals between limits to find exact numerical values

SPEC 8.3 • INTEGRATION

Indefinite vs Definite Integrals

Indefinite Integral

\int f (x) d x = F (x) + c

Gives a family of functions

Includes constant of integration + c

Definite Integral

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

Gives a numerical value

No constant of integration needed

Key Notation

In

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

:

a = lower limit (bottom number)

b = upper limit (top number)

We evaluate from x = a to x = b

The Fundamental Theorem of Calculus

Key Result

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

Where F(x) is the antiderivative (indefinite integral) of f(x)

Method

Step 1: Find the indefinite integral F(x) — don't include + c

Step 2: Write with square brackets and limits: [F(x)]ₐᵇ

Step 3: Substitute upper limit b into F(x) to get F(b)

Step 4: Substitute lower limit a into F(x) to get F(a)

Step 5: Calculate F(b) − F(a)

Critical

Always subtract in the correct order: F(upper) − F(lower), never the other way around!

Worked Example — Polynomial01

Evaluate $\int_{1}^{3} (3 x^{2} - 2 x) d x$

•

1. Find the indefinite integral:

\int (3 x^{2} - 2 x) d x = x^{3} - x^{2}

•

2. Write with limits:

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

•

3. Substitute upper limit (x = 3):

F (3) = 3^{3} - 3^{2} = 27 - 9 = 18

•

4. Substitute lower limit (x = 1):

F (1) = 1^{3} - 1^{2} = 1 - 1 = 0

•

5. Calculate F(3) − F(1):

18 - 0 = 18

Worked Example — Fractional Powers02

Evaluate $\int_{1}^{4} (x + \frac{1}{x}) d x$

•

1. Rewrite and integrate:

\int (x + x^{- \frac{1}{2}}) d x = \frac{x ^{2}}{2} + 2 x^{\frac{1}{2}} = \frac{x ^{2}}{2} + 2 x

•

2. Write with limits:

[\frac{x ^{2}}{2} + 2 x]_{1}^{4}

•

3. Substitute x = 4:

F (4) = \frac{16}{2} + 24 = 8 + 4 = 12

•

4. Substitute x = 1:

F (1) = \frac{1}{2} + 21 = \frac{1}{2} + 2 = \frac{5}{2}

•

5. Calculate:

12 - \frac{5}{2} = \frac{24}{2} - \frac{5}{2} = \frac{19}{2}

Worked Example — Exponential03

Evaluate $\int_{0}^{2} e^{3 x} d x$ , giving your answer in terms of e.

•

1. Integrate:

\int e^{3 x} d x = \frac{1}{3} e^{3 x}

•

2. Write with limits:

[\frac{1}{3} e^{3 x}]_{0}^{2}

•

3. Substitute x = 2:

F (2) = \frac{1}{3} e^{6}

•

4. Substitute x = 0:

F (0) = \frac{1}{3} e^{0} = \frac{1}{3} (1) = \frac{1}{3}

•

5. Calculate:

\frac{1}{3} e^{6} - \frac{1}{3} = \frac{1}{3} (e^{6} - 1)

Worked Example — Trigonometric04

Evaluate $\int_{0}^{\frac{π}{3}} cos 2 x d x$ , giving an exact answer.

•

1. Integrate:

\int cos 2 x d x = \frac{1}{2} sin 2 x

•

2. Write with limits:

[\frac{1}{2} sin 2 x]_{0}^{\frac{π}{3}}

•

3. Substitute x = π/3:

F (\frac{π}{3}) = \frac{1}{2} sin (\frac{2 π}{3}) = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}

•

4. Substitute x = 0:

F (0) = \frac{1}{2} sin (0) = 0

•

5. Calculate:

\frac{3}{4} - 0 = \frac{3}{4}

Worked Example — Logarithmic05

Evaluate $\int_{1}^{e} \frac{3}{x} d x$

•

1. Integrate:

\int \frac{3}{x} d x = 3 ln ∣ x ∣

•

2. Write with limits:

[3 ln ∣ x ∣]_{1}^{e}

•

3. Substitute x = e:

F (e) = 3 ln (e) = 3 \times 1 = 3

•

4. Substitute x = 1:

F (1) = 3 ln (1) = 3 \times 0 = 0

•

5. Calculate:

3 - 0 = 3

Worked Example — Combined Trig06

Evaluate $\int_{0}^{\frac{π}{4}} (sin x + cos x) d x$

•

1. Integrate:

\int (sin x + cos x) d x = - cos x + sin x

•

2. Write with limits:

[- cos x + sin x]_{0}^{\frac{π}{4}}

•

3. Substitute x = π/4:

F (\frac{π}{4}) = - cos \frac{π}{4} + sin \frac{π}{4} = - \frac{2}{2} + \frac{2}{2} = 0

•

4. Substitute x = 0:

F (0) = - cos (0) + sin (0) = - 1 + 0 = - 1

•

5. Calculate:

0 - (- 1) = 1

Properties of Definite Integrals

Key Properties

PropertyFormulaReversing Limits

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

Splitting Intervals

\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x

Constant Multiple

\int_{a}^{b} k \cdot f (x) d x = k \cdot \int_{a}^{b} f (x) d x

Sum/Difference

\int_{a}^{b} [f (x) \pm g (x)] d x = \int_{a}^{b} f (x) d x \pm \int_{a}^{b} g (x) d x

Using Properties

These properties help simplify complex integrals or find unknown integrals from given information.

Practice

Question 1: Evaluate

\int_{2}^{5} (x^{2} - 4 x + 3) d x

Step 1: Integrate:

\frac{x ^{3}}{3} - 2 x^{2} + 3 x

Step 2: Apply limits:

[\frac{x ^{3}}{3} - 2 x^{2} + 3 x]_{2}^{5}

Step 3: F(5) =

\frac{125}{3} - 50 + 15 = \frac{125}{3} - 35 = \frac{125 - 105}{3} = \frac{20}{3}

Step 4: F(2) =

\frac{8}{3} - 8 + 6 = \frac{8}{3} - 2 = \frac{8 - 6}{3} = \frac{2}{3}

Step 5:

\frac{20}{3} - \frac{2}{3} = 6

Practice

Question 2: Evaluate

\int_{0}^{\frac{π}{2}} (3 cos x - 2 sin x) d x

Step 1: Integrate:

3 sin x + 2 cos x

Step 2: Apply limits:

[3 sin x + 2 cos x]_{0}^{\frac{π}{2}}

Step 3: F(π/2) =

3 sin \frac{π}{2} + 2 cos \frac{π}{2} = 3 (1) + 2 (0) = 3

Step 4: F(0) =

3 sin (0) + 2 cos (0) = 0 + 2 = 2

Step 5:

3 - 2 = 1

Practice

Question 3: Evaluate

\int_{1}^{4} (2 e^{x} + \frac{3}{x}) d x

, giving your answer in terms of e and ln.

Step 1: Integrate:

2 e^{x} + 3 ln ∣ x ∣

Step 2: Apply limits:

[2 e^{x} + 3 ln ∣ x ∣]_{1}^{4}

Step 3: F(4) =

2 e^{4} + 3 ln 4

Step 4: F(1) =

2 e^{1} + 3 ln 1 = 2 e + 0 = 2 e

Step 5:

(2 e^{4} + 3 ln 4) - 2 e = 2 e^{4} - 2e + 3 ln 4

Or equivalently:

2 (e^{4} - e) + 3 ln 4

or

2 (e^{4} - e) + 6 ln 2

Summary

Key Formulae

Definite IntegralResult

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

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

\int_{a}^{b} x^{n} d x

[\frac{x ^{n + 1}}{n + 1}]_{a}^{b}

\int_{a}^{b} e^{k x} d x

[\frac{e ^{k x}}{k}]_{a}^{b}

\int_{a}^{b} \frac{1}{x} d x

ln \frac{b}{a}

(same sign limits)

Exam Tips

Always subtract: F(upper) − F(lower)

No "+ c" in definite integral answers

Give exact answers unless told otherwise

A negative result is valid — it indicates signed area