Indefinite and definite integration

IB Mathematics: Analysis and Approaches HL· 5.1, 5.2· 10 min read

1. Indefinite Integrals and Antiderivatives★★☆☆☆⏱ 15 min

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📘 Definition

Indefinite Integral

\int f(x) dx = F(x) + C

The collection of all antiderivatives of a function $f (x)$ is called the indefinite integral of $f (x)$ , where $F^{'} (x) = f (x)$ and $C$ is the arbitrary constant of integration.

Example:

$\int 2 x d x = x^{2} + C$

An indefinite integral has no fixed upper or lower limits, so it represents an infinite family of functions that differ only by a constant vertical shift. Any two antiderivatives of the same function will always differ by exactly this constant.

📐 Worked Example

Find the indefinite integral of $f (x) = 3 x^{2} + 2 sin x$

1
Recall basic integration rules: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ for $n \neq = - 1$ , and $\int sin x d x = - cos x + C$ . Apply linearity of integration:
2
$\int (3x^2 + 2\sin x) dx = 3 \cdot \frac{x^3}{3} + 2(-\cos x) + C$
3
Simplify to get the final general result:
4
$x^3 - 2\cos x + C$

Exam tip:

You will lose 1 mark in the exam for omitting the constant of integration $C$ in an indefinite integral answer. Always add it.

2. Definite Integrals and Net Signed Area★★☆☆☆⏱ 15 min

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📘 Definition

Definite Integral

\int_a^b f(x) dx

For a continuous function $f (x)$ on the interval $[a, b]$ , the definite integral is equal to the net signed area bounded by $y = f (x)$ , the x-axis, $x = a$ and $x = b$ . Areas above the x-axis are positive, areas below are negative.

Example:

$\int_{0}^{2} 2 x d x = 4$ , which equals the area of the triangle under the line between 0 and 2.

It is critical to distinguish between net signed area (the value of the definite integral) and total geometric area. If a question asks for total area between a curve and the x-axis, you must split the integral at every x-intercept and add the absolute value of each integral segment.

📐 Worked Example

Calculate $\int_{- 1}^{1} x d x$ using area interpretation.

1
Sketch $y = x$ over $[- 1, 1]$ . The graph forms two congruent right triangles: one above the x-axis from $x = 0$ to $x = 1$ , and one below from $x = - 1$ to $x = 0$ .
2
Calculate the signed area of each triangle: the upper triangle has area $\frac{1}{2} (1) (1) = 0.5$ , the lower triangle has signed area $- 0.5$ .
3
Add the signed areas to get the final result:
4
$\int_{-1}^1 x dx = 0.5 - 0.5 = 0$

3. Fundamental Theorem of Calculus★★★☆☆⏱ 20 min

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The Fundamental Theorem of Calculus (FTC) unites the two concepts of integration (antiderivatives and area) and gives a simple method to evaluate definite integrals without manual area calculation.

📘 Definition

Fundamental Theorem of Calculus (Part 2)

If $f (x)$ is continuous on $[a, b]$ and $F (x)$ is any antiderivative of $f (x)$ (meaning $F^{'} (x) = f (x)$ ), then: $\int_{a}^{b} f (x) d x = F (b) - F (a)$

📐 Worked Example

Evaluate $\int_{1}^{3} (2 x + e^{x}) d x$ using the FTC.

1
First find the general antiderivative (indefinite integral):
2
$\int (2x + e^x) dx = x^2 + e^x + C$
3
Ignore the constant of integration (it cancels out) and evaluate at the upper limit $b = 3$ and lower limit $a = 1$ :
4
$F(3) = 3^2 + e^3 = 9 + e^3, \quad F(1) = 1^2 + e^1 = 1 + e$
5
Subtract lower limit from upper limit to get the final result:
6
$\int_1^3 (2x + e^x) dx = (9 + e^3) - (1 + e) = 8 + e^3 - e$

Exam tip:

Always evaluate upper limit minus lower limit. Swapping the order gives the negative of the correct answer and will cost you marks.

4. Common Pitfalls

Wrong move:

Omitting the constant of integration $C$ in an indefinite integral answer

Why:

IB examiners explicitly penalize missing $C$ , as it is required to represent the full family of antiderivatives

Correct move:

Always add $+ C$ at the end of any indefinite integral result

Wrong move:

Calculating $F (a) - F (b)$ instead of $F (b) - F (a)$ for definite integrals

Why:

Swapping the order of limits gives the incorrect sign for the final answer

Correct move:

Memorize: upper limit (second number) minus lower limit (first number)

Wrong move:

Treating all area as positive when calculating a definite integral of a function that crosses the x-axis

Why:

The definite integral returns net signed area, not total geometric area

Correct move:

Keep signs for definite integrals; split the integral and add absolute values if asked for total area

Wrong move:

Forgetting that integration is the inverse of differentiation when checking antiderivatives

Why:

Differentiating your result is the fastest way to confirm you have the correct antiderivative

Correct move:

Always differentiate your final antiderivative to check it matches the original integrand

5. Quick Reference Cheatsheet

Concept	Notation	Key Rule
Indefinite Integral	$\int f (x) d x$	Family of antiderivatives: $F (x) + C$
Definite Integral	$\int_{a}^{b} f (x) d x$	Net signed area between $x = a$ and $x = b$
FTC Evaluation	$\int_{a}^{b} f (x) d x$	$F (b) - F (a)$ , where $F^{'} (x) = f (x)$
Constant of Integration	$C$	Required only for indefinite integrals

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 1
Evaluate definite integral of trigonometric function
2022 · 2
Find general indefinite integral of polynomial

What's Next

Mastering the core definitions and rules of indefinite and definite integration is the essential foundation for all upcoming integration topics in IB AA HL. All advanced techniques including integration by substitution, integration by parts, partial fraction integration, and applications of integration to find areas between curves and volumes of revolution rely on your ability to correctly apply these core concepts. Regular practice distinguishing between indefinite and definite integrals and applying the Fundamental Theorem of Calculus will make all more advanced integration topics much easier to master.