IB Math AA SL Topic 5.5: Introduction to integration | Notes & Flashcards | Aimnova

IB Math AA SL — Introduction to integration

Topic 5.5 of IB Mathematics: Analysis and Approaches covers Introduction to integration, which is part of Unit 5: Calculus. Students explore key concepts including Antidifferentiation, Area under a curve. A strong understanding of introduction to integration is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Introduction to integration

Key Idea: Integration is differentiation in reverse — it recovers a function from its rate of change and measures the area under a curve. On Paper 1 you integrate by hand, so the power rule and the + C habit are everything.

∫ The power rule for integration

\int x^{n}\,dx = \frac{x^{\,n+1}}{n+1} + C \quad (n \ne -1)

$n$: the power — add 1 to it, then divide by the new power
$C$: the constant of integration — always there on an indefinite integral

Integration undoes differentiation: add 1 to the power, divide by the new power, then add + C. Integrate a polynomial term by term with one + C at the end. A constant like 5 integrates to 5x (think 5 = 5x⁰).

🧩 Indefinite vs definite

\text{Area} = \int_a^b f(x)\,dx \qquad (f \ge 0)

$a, b$: the lower and upper limits (the x-values bounding the region)

Tip: Turn roots and fractions into powers before integrating: $\sqrt{x} = x¹/²$ and $1/x² = x⁻²$. Dividing by 3/2 means × 2/3. The rule excludes n = −1 (that case is logs, Topic 5.x).

✏️ IB-style worked examples

IB-style question — integrate, then find f from a point

(a) Find ∫(9x² − 4x + 7) dx. (b) Find ∫(√x + 1/x²) dx. (c) A curve has f′(x) = 6x² − 2 and passes through (1, 5). Find f(x).

Step by step:

(a) Integrate term by term, one + C at the end.
$\frac{9x^{3}}{3} - \frac{4x^{2}}{2} + 7x + C = 3x^{3} - 2x^{2} + 7x + C$
(b) Rewrite as powers first: √x = x¹/², 1/x² = x⁻².
$\int (x^{1/2} + x^{-2})\,dx = \frac{2}{3}x^{3/2} - \frac{1}{x} + C$
(c) Integrate f′, then use f(1) = 5 to find C.
$f(x) = 2x^{3} - 2x + C,\quad 2 - 2 + C = 5 \Rightarrow C = 5$

Final answer:

(a) 3x³ − 2x² + 7x + C (b) (2/3)x³/² − 1/x + C (c) f(x) = 2x³ − 2x + 5

IB-style question — evaluate, then find a limit from a given area

(a) Evaluate ∫₁⁴ 2x dx. (b) The area under y = 2x from x = 0 to x = k is 16 (k > 0). Find k.

Step by step:

(a) Integrate (no + C), then do F(top) − F(bottom).
$[x^{2}]_1^{4} = 4^{2} - 1^{2} = 16 - 1 = 15$
(b) Set the definite integral equal to the area.
$\int_0^{k} 2x\,dx = [x^{2}]_0^{k} = k^{2} = 16$
(b) Solve and keep the sensible (positive) root.
$k = 4$

Final answer:

(a) 15 (b) k = 4

Important: An indefinite integral always needs + C — leaving it off loses a mark. And when you're given a point, the work isn't done until you've used it to find C — don't leave the answer as … + C.

Tap each card to reveal the answer.

Exam Tips

Power rule: add 1 to the power, divide by the new power, + C.
Indefinite → + C (and find C from a point if one is given); definite → a number, no + C.
Rewrite roots and fractions as powers (√x = x¹/², 1/xⁿ = x⁻ⁿ) before integrating.
Definite integral = F(top) − F(bottom); reversing the limits flips the sign.
Area given? Set the definite integral equal to it and solve for the unknown limit (keep the sensible root).

IB Math AA SL — Introduction to integration

Key concepts in Introduction to integration

Key Idea: Integration is differentiation in reverse — it recovers a function from its rate of change and measures the area under a curve. On Paper 1 you integrate by hand, so the power rule and the + C habit are everything.

∫ The power rule for integration

\int x^{n}\,dx = \frac{x^{\,n+1}}{n+1} + C \quad (n \ne -1)

$n$: the power — add 1 to it, then divide by the new power
$C$: the constant of integration — always there on an indefinite integral

Integration undoes differentiation: add 1 to the power, divide by the new power, then add + C. Integrate a polynomial term by term with one + C at the end. A constant like 5 integrates to 5x (think 5 = 5x⁰).

🧩 Indefinite vs definite

\text{Area} = \int_a^b f(x)\,dx \qquad (f \ge 0)

$a, b$: the lower and upper limits (the x-values bounding the region)

Tip: Turn roots and fractions into powers before integrating: $\sqrt{x} = x¹/²$ and $1/x² = x⁻²$. Dividing by 3/2 means × 2/3. The rule excludes n = −1 (that case is logs, Topic 5.x).

✏️ IB-style worked examples

IB-style question — integrate, then find f from a point

(a) Find ∫(9x² − 4x + 7) dx. (b) Find ∫(√x + 1/x²) dx. (c) A curve has f′(x) = 6x² − 2 and passes through (1, 5). Find f(x).

Step by step:

(a) Integrate term by term, one + C at the end.
$\frac{9x^{3}}{3} - \frac{4x^{2}}{2} + 7x + C = 3x^{3} - 2x^{2} + 7x + C$
(b) Rewrite as powers first: √x = x¹/², 1/x² = x⁻².
$\int (x^{1/2} + x^{-2})\,dx = \frac{2}{3}x^{3/2} - \frac{1}{x} + C$
(c) Integrate f′, then use f(1) = 5 to find C.
$f(x) = 2x^{3} - 2x + C,\quad 2 - 2 + C = 5 \Rightarrow C = 5$

Final answer:

(a) 3x³ − 2x² + 7x + C (b) (2/3)x³/² − 1/x + C (c) f(x) = 2x³ − 2x + 5

IB-style question — evaluate, then find a limit from a given area

(a) Evaluate ∫₁⁴ 2x dx. (b) The area under y = 2x from x = 0 to x = k is 16 (k > 0). Find k.

Step by step:

(a) Integrate (no + C), then do F(top) − F(bottom).
$[x^{2}]_1^{4} = 4^{2} - 1^{2} = 16 - 1 = 15$
(b) Set the definite integral equal to the area.
$\int_0^{k} 2x\,dx = [x^{2}]_0^{k} = k^{2} = 16$
(b) Solve and keep the sensible (positive) root.
$k = 4$

Final answer:

(a) 15 (b) k = 4

Important: An indefinite integral always needs + C — leaving it off loses a mark. And when you're given a point, the work isn't done until you've used it to find C — don't leave the answer as … + C.

Tap each card to reveal the answer.

Exam Tips

Power rule: add 1 to the power, divide by the new power, + C.
Indefinite → + C (and find C from a point if one is given); definite → a number, no + C.
Rewrite roots and fractions as powers (√x = x¹/², 1/xⁿ = x⁻ⁿ) before integrating.
Definite integral = F(top) − F(bottom); reversing the limits flips the sign.
Area given? Set the definite integral equal to it and solve for the unknown limit (keep the sensible root).

IB Math AA SL — Introduction to integration

Key concepts in Introduction to integration

∫ The power rule for integration

🧩 Indefinite vs definite

✏️ IB-style worked examples

IB-style question — integrate, then find f from a point

IB-style question — evaluate, then find a limit from a given area

Exam Tips

What you'll learn in Topic 5.5

Study resources — 5.5 Introduction to integration

Antidifferentiation

Area under a curve

Ready to study Introduction to integration?

Ready to practice?

IB Math AA SL — Introduction to integration

Key concepts in Introduction to integration

∫ The power rule for integration

🧩 Indefinite vs definite

✏️ IB-style worked examples

IB-style question — integrate, then find f from a point

IB-style question — evaluate, then find a limit from a given area

Exam Tips

What you'll learn in Topic 5.5

Study resources — 5.5 Introduction to integration

Antidifferentiation

Area under a curve

Ready to study Introduction to integration?

Ready to practice?