Mathematics · Lesson 24

Integration (JAMB calculus)

JAMB

What the syllabus asks

JAMB: integration of explicit algebraic and simple trigonometric functions; area under the curve (applications restricted to areas bounded by curves and the axes)

What you will learn

Explain integration as the reverse of differentiation.
Integrate polynomials and simple powers, including negative and fractional powers.
Find the equation of a curve from its gradient function and a point.
Evaluate definite integrals correctly.
Find areas bounded by a curve, the x-axis and given lines, including regions below the axis.
Integrate simple trigonometric functions.
Apply integration to motion (acceleration, velocity, displacement).

Key points

Integration reverses differentiation; add + C to indefinite integrals.
Power rule: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1; integrate term by term.
Rewrite roots and fractions as powers: √x = x^½, 1/x² = x⁻².
Expand products and split single-term fractions before integrating; ∫(ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n + 1)] + C.
Use a given point to find C and the exact curve.
Definite integral: F(b) − F(a); no + C.
Area under a curve: ∫ₐᵇ y dx; find limits from y = 0 if not given.
Regions below the axis give negative integrals: split at roots and add absolute values.
Area between curves: ∫(top − bottom) dx between intersection points.
∫cos x dx = sin x + C, ∫sin x dx = −cos x + C (radians); v = ∫a dt, s = ∫v dt.

Start here: Integration as reverse differentiation

In the lesson “Differentiation” you learnt to differentiate: given a function, find its gradient function. Integration asks the reverse question: given the gradient function, what was the original function?

If dy/dx = 3x², what was y? We know d/dx(x³) = 3x², so y = x³ is one answer. But d/dx(x³ + 5) = 3x² too, and so does d/dx(x³ − 100). Any constant disappears when we differentiate, so we must add an unknown constant:

If F′(x) = f(x), then ∫f(x) dx = F(x) + C, where C is the constant of integration. The ∫ sign means "integrate", and "dx" tells us that x is the variable.

Geometrically, all the curves y = x³ + C are the same shape, shifted up or down; they all have the same gradient at each x.

Worked example 24.1

By thinking backwards, find (a) ∫2x dx, (b) ∫5 dx, (c) ∫4x³ dx.

(a) d/dx(x²) = 2x → x² + C.
(b) d/dx(5x) = 5 → 5x + C.
(c) d/dx(x⁴) = 4x³ → x⁴ + C.

x² + C; 5x + C; x⁴ + C

Why does this matter in real life? Very often we know a rate but want a total. A water company knows how fast water flows into a tank each minute and wants the total volume; a doctor knows how fast a drug enters the blood and wants the total dose absorbed; an engineer knows a car’s speed at every moment and wants the distance covered. In each case, integration turns the rate into the total. That is why integration, together with differentiation, is one of the two pillars of calculus.

The notation was introduced by the German mathematician Leibniz in the 1670s: the long S, ∫, stands for “summa” (sum), because, as you will see in Section 24.7, an integral is really the sum of infinitely many tiny pieces.

💡 Exam tip: you can always check an integral by differentiating your answer: you should get back the integrand.

⚠️ Common mistake: leaving out "+ C" in an indefinite integral. Examiners deduct marks for it.

Try it yourself. Find ∫6x² dx by reverse thinking.

Show solution

d/dx(2x³) = 6x² → 2x³ + C.

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Practice questions

1. Find ∫(4x³ − 6x + 5) dx.

Show worked answer

x⁴ − 3x² + 5x + C.

2. Find ∫(2/x³ + √x) dx.

Show worked answer

−1/x² + (2/3)x^(3/2) + C.

3. A curve has dy/dx = 2x + 3 and passes through (1, 6). Find its equation.

Show worked answer

y = x² + 3x + 2.

4. Evaluate ∫₀² (x³ + 1) dx.

Show worked answer

[x⁴/4 + x]₀² = 6.

5. Find the area enclosed by y = 6x − x² and the x-axis.

Show worked answer

∫₀⁶ = 108 − 72 = 36 square units.

Key words

Integration: The process of finding a function whose derivative is given.
Antiderivative: A function F with F′(x) = f(x).
Indefinite integral: ∫f(x) dx = F(x) + C, a family of functions.
Constant of integration C: The unknown constant added in an indefinite integral.
Integrand: The function being integrated.
Definite integral: ∫ₐᵇ f(x) dx = F(b) − F(a), a number.
Limits of integration: The values a (lower) and b (upper).
Area under a curve: Area between the curve, the x-axis and two vertical lines.
Boundary condition: A known point used to find C.
Displacement: Position relative to a starting point; s = ∫v dt.
Velocity: Rate of change of displacement; v = ∫a dt.
Radian: The angle unit used in calculus; π rad = 180°.

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