Integration

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• Mathematics
• Further Education (Key Stage 5)

Integration: reversing change

Open with the big idea that integration undoes differentiation and lets us recover functions and measure total accumulation. Keep the title slide brief and purposeful for a Year 12 resit student.

Learning goal

I can integrate standard functions, include the constant of integration correctly, and solve Edexcel-style problems involving indefinite and definite integrals.

What words or symbols do you link with integration?

Use this to surface prior knowledge and anxieties. Likely responses: area, reverse differentiation, ∫, + c, antiderivative, ln x, limits. Pick up any misconceptions for attention later.

Integration and differentiation

Integration is the reverse of differentiation.

If

d/dx (xⁿ) = nxⁿ⁻¹,

then reversing this gives a rule for finding antiderivatives.

When we integrate, we are finding a family of functions whose derivative is the given expression.

Stress the word antiderivative. Students often see integration as a new process rather than an inverse process. Link immediately to known differentiation rules.

The power rule for integration

For n ≠ -1,

∫xⁿ dx = xⁿ⁺¹/(n+1) + c

Key idea:

• increase the power by 1

• divide by the new power

• add + c

Example:

∫x³ dx = x⁴/4 + c

Model careful notation: the integral sign, dx, and + c every time for indefinite integrals. Mention that n = -1 is excluded because division by zero would occur.

Why + c matters

Different functions can have the same derivative.

Example:

• d/dx (x² + 5) = 2x

• d/dx (x² - 3) = 2x

So

∫2x dx = x² + c

The constant of integration represents all possible vertical shifts.

A common exam error is omitting + c on indefinite integrals. Emphasise that it is not optional and costs marks in Edexcel questions.

Edexcel-style basics

Find:

• ∫6x⁵ dx

• ∫(4x - 7) dx

• ∫(3/x²) dx

Useful rewrite:

3/x² = 3x⁻²

Use this as live worked practice. Expected answers: x⁶ + c, 2x² - 7x + c, -3x⁻¹ + c or -3/x + c. Check whether the student rewrites fractional forms correctly before integrating.

Linearity of integration

We can integrate term by term.

∫(axⁿ + bxᵐ + k) dx = a∫xⁿ dx + b∫xᵐ dx + ∫k dx

Example:

∫(5x² - 4x + 9) dx = 5x³/3 - 2x² + 9x + c

This slide supports algebraic fluency. Students often make arithmetic slips with coefficients; encourage slow, line-by-line working.

Which statement is correct for an indefinite integral?

A

Add + c

B

Use limits

C

Find F(b) - F(a)

D

Give one number

Checks the core distinction built in slides 2 to 6: indefinite integrals give a family of functions, so + c is required.

Complete the integration summary

For n ≠ -1, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + c. When integrating 2x, the answer is x² + c because different vertical shifts have the same derivative. We can also integrate term by term for expressions such as 5x² - 4x + 9.

∫xⁿ dx = xⁿ⁺¹/(n + 1) + c

2x

term by term

F(b) - F(a)

ln x + c

multiply powers

The blanks target the power rule, the need for + c, and linearity. Prompt the student to read the full paragraph aloud after completing it.

Negative and fractional powers

The power rule still works for fractional and negative powers, except x⁻¹.

Examples:

• ∫x¹ᐟ² dx = 2x³ᐟ²/3 + c

• ∫x⁻³ dx = x⁻²/(-2) + c = -1/(2x²) + c

Always rewrite roots and reciprocals as powers when possible.

Pause on notation. Read x¹ᐟ² as square root x and x⁻³ as 1/x³. This is often where resit students lose confidence, so keep examples simple and clean.

The special case ln x

∫1/x dx = ln x + c

This is the case missed out by the power rule because n = -1.

Use ln|x| if you want full mathematical generality, but Edexcel A-level contexts often present this in domains where x > 0. Match the course specification and question style.

Mixed algebraic forms

Before integrating, rewrite where needed.

Example:

∫(2√x + 3/x - 4x²) dx

becomes

∫(2x¹ᐟ² + 3x⁻¹ - 4x²) dx

Then integrate each term separately.

Work this through orally or on paper: 4x³ᐟ²/3 + 3ln x - 4x³/3 + c. Students often forget that only the x⁻¹ term gives ln x.

Edexcel-style fluency

Find:

• ∫(x¹ᐟ² + x⁻¹) dx

• ∫(4/x³ - 5√x) dx

• ∫(6 - 2/x) dx

Aim for accurate algebra and correct use of ln x.

Expected answers: 2x³ᐟ²/3 + ln x + c; -2/x² - 10x³ᐟ²/3 + c; 6x - 2ln x + c. Check whether the student leaves answers in acceptable equivalent forms.

What is ∫1/x dx?

A

x⁰/0 + c

B

1/x² + c

C

ln x + c

D

x⁻²/(-2) + c

Distractors reflect common misuse of the power rule and confusion with x⁻². Reinforce that x⁻¹ is the special case.

Complete the mixed-forms method

Before integrating mixed algebraic forms, rewrite roots and reciprocals as powers. For example, 2√x becomes 2x¹ᐟ² and 3/x becomes 3x⁻¹. Then integrate each term separately, remembering that only the x⁻¹ term gives ln x.

rewrite

2x¹ᐟ²

3x⁻¹

x⁻¹

expand

3x

x⁻²

x²

Echoes slides 7 to 10 exactly: rewrite first, then choose between power rule and ln x appropriately.

Why does 1/x need a different rule?

Expected elements: identify that 1/x is the special case; explain it corresponds to x⁻¹ where the power rule would require division by zero; mention that its integral is ln x + c.

Finding a particular function

Sometimes integration gives a family of functions, then extra information fixes c.

Example:

Given f′(x) = 6x + 4 and f(1) = 3

Integrate:

f(x) = 3x² + 4x + c

Use f(1) = 3 to find c.

Let the student substitute: 3 + 4 + c = 3, so c = -4. Then f(x) = 3x² + 4x - 4. Emphasise the two-step structure often seen in exams.

Using a point on the curve

A gradient function and one point can determine the whole curve.

Example:

The curve has gradient dy/dx = 9x² - 2 and passes through (0, 5).

Integrate:

y = 3x³ - 2x + c

Then use the point to find c.

Expected constant is c = 5. Highlight vocabulary: passes through (0, 5) means when x = 0, y = 5.

Exam technique with constants

Good structure for these questions:

• write the integrated function first

• include + c immediately

• substitute the given coordinates carefully

• solve for c

• write the final function clearly

This is about method marks. Edexcel questions reward organised working even if arithmetic goes wrong later.

Edexcel-style function question

A curve has gradient function

dy/dx = 4x³ - 6x

and passes through (1, 2).

Find an equation of the curve.

Expected answer: y = x⁴ - 3x² + c, then 1 - 3 + c = 2 so c = 4, giving y = x⁴ - 3x² + 4. Encourage full statement at the end.

After integrating dy/dx, what should you do next in a curve question?

A

Add limits

B

Differentiate again

C

Substitute the point

D

Set y = 0

Checks the exam structure from slides 11 to 14: integrate, include + c, then use the given point or value to find c.

Complete the curve-finding method

When a question gives a gradient function, first integrate to find a family of functions. Include + c immediately. Then substitute the given point, solve for c, and write the final equation clearly.

integrate

+ c

substitute

c

differentiate

F(b) - F(a)

expand

x-axis

This anchors the method marks and sequence stressed on slide 13.

From antiderivative to area

A definite integral gives the signed area between a curve and the x-axis over an interval.

∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) - F(a)

where F′(x) = f(x).

This is why finding an antiderivative matters.

Introduce the notation [F(x)]ₐᵇ carefully. Signed area means parts below the x-axis count as negative.

Evaluating definite integrals

Example:

∫₁³ 2x dx = [x²]₁³ = 3² - 1² = 8

Method:

• integrate first

• use brackets

• substitute the upper limit

• subtract the lower limit

Students often reverse the subtraction or forget brackets when substituting negative values. Reinforce the exact sequence every time.

Signed area

If the curve is below the x-axis, the definite integral is negative.

Example:

∫₀² (-3) dx = [-3x]₀² = -6

So a definite integral is not always the same as total area.

This distinction matters later in area questions. Keep language precise: definite integral gives signed area, while total area requires positive values.

Area under a positive curve

When f(x) ≥ 0 on the interval,

area = ∫ₐᵇ f(x) dx

Example:

For y = x² from x = 0 to x = 2,

area = ∫₀² x² dx.

You can quickly finish this orally: [x³/3]₀² = 8/3. Keep the visual idea linked to the formal notation.

Edexcel-style definite integrals

Evaluate:

• ∫₀⁴ (x + 1) dx

• ∫₁² 3x² dx

• ∫₋₁¹ 2x dx

Look for symmetry and careful substitution.

Expected answers: 12, 7, 0. The third example is useful because the integrand is odd and the interval is symmetric, but students should still be able to verify by calculation.

What does a definite integral give?

A

A gradient function

B

A signed area

C

A vertical shift

D

A family of curves

Built directly from slides 15 to 19. Use follow-up discussion to distinguish signed area from total area.

Complete the definite-integral method

A definite integral is evaluated by finding an antiderivative first. Then use F(b) - F(a) with the upper limit before the lower limit. If the curve is below the x-axis, the result is negative because the integral gives signed area.

antiderivative

F(b) - F(a)

x-axis

negative

derivative

+ c

positive

root

Useful for exposing two classic errors: reversing the subtraction and confusing signed area with total area.

Area between a curve and the x-axis

To find area, first check where the curve is above or below the x-axis.

If a curve crosses the axis, split the integral at the roots.

Then add the positive areas of each region.

This is the conceptual bridge to standard A-level area questions. Do not overload with too many cases; one clean example is enough.

A split-area example

For y = x² - 1 between x = -2 and x = 2:

• roots at x = -1 and x = 1

• below the axis for -1 < x < 1

• above the axis elsewhere

So the total area must be split into separate integrals.

You do not need to complete all arithmetic on the slide. The key exam skill is spotting that one integral over the whole interval would not give the total area.

Common exam errors

• forgetting + c on indefinite integrals

• using the power rule for 1/x

• not rewriting roots or reciprocals as powers

• reversing F(b) - F(a)

• treating signed area as total area

Make this a quick consolidation checkpoint. These are high-frequency Edexcel errors for integration questions.

Edexcel-style mixed practice

1. Find ∫(7x⁶ - 2/x) dx

1. Given dy/dx = 5x⁴ + 3x² and the curve passes through (0, -2), find the equation of the curve.

1. Evaluate ∫₀² (3x² - 4) dx

Expected answers: x⁷ - 2ln x + c; y = x⁵ + x³ - 2; 0. This final set lets you assess rule selection, constants, and definite-integral method in one place.

If a curve crosses the x-axis, how do you find the total area?

A

Use one integral only

B

Double the answer

C

Split at the roots

D

Ignore negative parts

Checks the conceptual move in slides 20 to 23. Students must distinguish total area from a single signed integral.

Why can one integral give the wrong total area?

Expected elements: one whole integral can cancel positive and negative regions; you must identify roots, split the interval, and add positive areas to get total area.

What is one integration idea you now feel clearer about?

Reflection only. Encourage the student to name a specific rule, method, or exam habit they improved.

Which part of today's lesson felt hardest?

Reflection only. Useful for planning follow-up revision before the resit.

What exam tip from this lesson will you use next time?

Reflection only. Prompt for a practical takeaway such as '+ c on indefinite integrals', 'split at roots', or 'upper minus lower'.

Glossary

integral - A mathematical expression representing an antiderivative or a signed area.

integration - The process of finding an integral.

antiderivative - A function whose derivative is the given function.

constant of integration - The constant c added to an indefinite integral.

indefinite integral - An integral with no limits, giving a family of functions.

definite integral - An integral with upper and lower limits, giving a numerical value.

signed area - Area counted as positive above the x-axis and negative below it.

Glossary

upper limit - The top value of the interval in a definite integral.

lower limit - The bottom value of the interval in a definite integral.

root - A value of x for which the function equals 0.

gradient function - The derivative, such as dy/dx, which gives the gradient at each point.

x-axis - The horizontal axis used when interpreting area under a curve.