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Module 3 · L10 of 15 ~35 min +95 XP available

Integration

Integration is the reverse of differentiation. If differentiation tells you the rate of change, integration recovers the total change from the rate. Like reconstructing a journey from the speedometer readings, integration accumulates small pieces into a whole.

Today's hook, If you know a car's velocity at every instant, can you recover the total distance travelled? If you know how fast a water tank fills at every moment, can you find the total volume? Integration does exactly this, it reverses differentiation to recover the whole from its rate.

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Orient to integration

Work through the visible teaching and complete each embedded check.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall, your gut answer first

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If differentiating $x^3$ gives $3x^2$, what operation would take $3x^2$ back to $x^3$? What do you think the reverse of differentiation might look like? What rule would you use?

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The two moves

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There are only two moves in integration. Master these and every polynomial integral is mechanical.

Move 1, Raise and divide: to integrate $x^n$, add 1 to the power and divide by the new power. This is the exact reverse of differentiation (which multiplied by the power and subtracted 1).

Move 2, Always add $+C$: the constant of integration is not optional. Any constant differentiates to zero, so every antiderivative is really a family of functions differing only by their constant.

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

Raise and divide

$\int x^n \, dx$: power goes from $n$ to $n+1$, divide by $n+1$. Opposite of “multiply by $n$, reduce power” in differentiation.

Always write $+C$

Forgetting $+C$ loses marks in the HSC every year. The $+C$ represents the entire family of antiderivatives, not just one.

Verify by differentiating

Differentiate your answer to check: you should recover the original integrand. This is a quick, reliable check.

What you will master

Know

Key facts

The power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
That $n \neq -1$ for this rule to apply
That $+C$ is always required for indefinite integrals

Understand

Concepts

Why integration is the reverse of differentiation
Why infinitely many antiderivatives exist for any function
How a known point on the curve pins down the value of $C$

Can do

Skills

Integrate polynomial functions term by term
Simplify expressions before integrating
Verify an integral by differentiating the result
Find a specific antiderivative given a point on the curve

Build the integration language

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Key terms

IntegrationThe process of finding a function from its derivative; the reverse of differentiation.

AntiderivativeA function $F(x)$ such that $F'(x) = f(x)$. Also called the indefinite integral.

Constant of integrationThe arbitrary constant $C$ added to an indefinite integral, representing the family of all antiderivatives.

Indefinite integralAn integral without limits: $\int f(x) \, dx = F(x) + C$.

Power rule (integration)$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.

IntegrandThe function being integrated; the expression after the integral sign and before $dx$.

Reverse differentiation

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Integration as the reverse of differentiation

core concept

If differentiation breaks a function down into its rate of change, integration builds it back up. The power rule for integration is:

$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1$$

Every indefinite integral includes the arbitrary constant $C$ because the derivative of any constant is zero. This means the function $F(x) + C$ differentiates back to $f(x)$ for any value of $C$, so there is a whole family of antiderivatives.

Integration is term by term

For a polynomial, integrate each term separately using the power rule, then combine:

$$\int (3x^2 + 4x - 5) \, dx = x^3 + 2x^2 - 5x + C$$

Constants behave simply

A constant $k$ integrates to $kx$, because $k = kx^0$ and the power rule gives $\frac{kx^1}{1} = kx$.

Real-world link. If $v(t)$ is velocity (the derivative of position), then integrating $v(t)$ recovers the position function $s(t)$. This is why integration is described as “accumulating” a quantity: you are undoing the differentiation that gave the rate of change.

Power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, raise power by 1, divide by new power; Constant: $\int k \, dx = kx + C$

Pause, copy the power rule $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ (raise power by 1, divide by new power) and the constant rule $\int k\,dx = kx + C$ into your book.

Quick check: True or false, the integral $\int x^3 \, dx = \frac{x^4}{4} + C$.

Apply the power rule

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Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · BASIC POWER RULE

Find $\int 3x^2 \, dx$.

$\displaystyle\int 3x^2 \, dx = 3 \cdot \frac{x^3}{3} + C$

Apply the power rule: raise the power from 2 to 3, divide by the new power 3. The coefficient 3 stays out front.

PROBLEM 2 · POLYNOMIAL INTEGRATION

Find $\int (4x^3 - 2x + 5) \, dx$.

$\displaystyle\int 4x^3 \, dx = \frac{4x^4}{4} = x^4$

Integrate the first term using the power rule.

PROBLEM 3 · SIMPLIFY FIRST

Find $\displaystyle\int \frac{x^4 + 3x^2}{x} \, dx$.

$\dfrac{x^4 + 3x^2}{x} = x^3 + 3x$

Simplify by dividing each term in the numerator by $x$ before integrating. You cannot integrate a fraction like this directly.

Quick check: Which of the following is the correct integral of $\int (2x + 3) \, dx$?

Avoid integration traps

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Common errors · the 3 traps that cost marks

Trap 01

Forgetting the constant of integration $+C$

Every indefinite integral must include $+C$. Without it, you are only finding one antiderivative instead of the complete family. This costs marks in HSC exams and examiners actively check for it.

Trap 02

Applying the power rule when $n = -1$

The power rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ does not work when $n = -1$ because you would divide by zero. The special case $\int \frac{1}{x} \, dx = \ln|x| + C$ is not in the Year 11 syllabus but you must recognise the restriction.

Trap 03

Forgetting to simplify before integrating

For $\int \frac{x^2 + 1}{x} \, dx$, you must expand to $\int \! \left(x + \frac{1}{x}\right) dx$ first. Trying to integrate the fraction directly leads to errors. Always simplify algebraically before applying the rule.

Odd one out: Three of the following are correct antiderivatives of $3x^2$. Which one is NOT?

Practise and reflect

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Quick-fire practice · 5 problems

Find $\int 5x^4 \, dx$.

Find $\int (2x + 3) \, dx$.

Find $\int (x^3 - 4x^2 + x) \, dx$.

Find $\displaystyle\int \frac{x^5 - 2x^3}{x^2} \, dx$.

If $f'(x) = 3x^2 - 2x$ and $f(1) = 4$, find $f(x)$.

Fill the blanks: drag each token into the matching blank.

add divide constant differentiate

To integrate $x^n$, ___ 1 to the power and ___ by the new power. Always include the ___ of integration $+C$. To verify your answer, ___ it to check you recover the original integrand.

Think aloud: Given $f'(x) = 6x^2 - 4x + 1$ and $f(2) = 10$, find $f(x)$. Explain every step.

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Revisit your thinking

Earlier you were asked: What operation takes $3x^2$ back to $x^3$? Integration does exactly this. The power rule, raise power by 1, divide by new power, reverses differentiation. The constant $C$ appears because many different functions can share the same derivative (e.g. $x^3$, $x^3 + 5$, $x^3 - 17$ all differentiate to $3x^2$).

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Complete the mixed practice

Complete the question-bank round and short-answer practice, then compare your responses with the comprehensive answers.

Multiple choice

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Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 4

Basic polynomial integration

Find $\int (4x^3 - 6x^2 + 2) \, dx$. 2 MARKS

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View comprehensive answer

$\int 4x^3 \, dx = x^4$ [0.5]

$\int -6x^2 \, dx = -2x^3$ [0.5]

$\int 2 \, dx = 2x$ [0.5]

Answer: $x^4 - 2x^3 + 2x + C$ [0.5]

ApplyBand 4

Integration after simplification

Find $\displaystyle\int \frac{2x^4 + 3x^2}{x} \, dx$. 3 MARKS

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View comprehensive answer

$\dfrac{2x^4 + 3x^2}{x} = 2x^3 + 3x$ [0.5]

$\int 2x^3 \, dx = \frac{x^4}{2}$ [0.5]

$\int 3x \, dx = \frac{3x^2}{2}$ [0.5]

Answer: $\dfrac{x^4}{2} + \dfrac{3x^2}{2} + C$ or $\dfrac{1}{2}(x^4 + 3x^2) + C$ [1.5]

ApplyBand 4

Find function from derivative and point

Given $f'(x) = 6x^2 - 4x + 1$ and $f(2) = 10$, find $f(x)$. 4 MARKS

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View comprehensive answer

$f(x) = \int (6x^2 - 4x + 1) \, dx = 2x^3 - 2x^2 + x + C$ [1.5]

$f(2) = 16 - 8 + 2 + C = 10$ [1]

$10 + C = 10$, so $C = 0$ [1]

$f(x) = 2x^3 - 2x^2 + x$ [0.5]

📖 Comprehensive answers (click to reveal)

Drill 1: $x^5 + C$

Drill 2: $x^2 + 3x + C$

Drill 3: $\frac{x^4}{4} - \frac{4x^3}{3} + \frac{x^2}{2} + C$

Drill 4: Simplify: $x^3 - 2x$. Integrate: $\frac{x^4}{4} - x^2 + C$

Drill 5: $f(x) = x^3 - x^2 + C$. $f(1) = 1 - 1 + C = 4 \implies C = 4$. So $f(x) = x^3 - x^2 + 4$.

Review and complete

Retrieve the central ideas, then mark the lesson complete or continue to the module quiz.

Retrieve the lesson before you leave

State the power rule for integration and its restriction.
Explain why an indefinite integral needs the constant of integration.
Describe how differentiating an answer verifies an antiderivative.

Answer from memory first, then return to the matching Learn checkpoint to check and correct your response.

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Module overview · Maths Advanced · Checkpoint 1