IntegrationRules & Examples
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∫ What is Integration?

Integration is the reverse of differentiation. If differentiating gives the rate of change, integrating accumulates — adding up infinitely many tiny slices to find area, distance, charge, work, and more.

Reverse of d/dx+c always for indefinite Area under a curves = ∫v dt
∫ 3x² dx = x³ + c All five curves share derivative 3x² — constant c only shifts the curve up or down xy -2 -1 1 2 -4 -2 2 4 y=x³−3 y=x³−1 y=x³ (c=0) y=x³+2 y=x³+4

🎯 Riemann Sum — drag to add more rectangles and watch the approximation improve

ℹ️ How to use: Drag the slider to change the number of rectangles. As n increases, the rectangle sum gets closer to the true area under the curve — this is the core idea of integration.

4
n=4 | Sum=6.6250 | True=6.6667 | Error=0.0417
💡 As n → ∞ rectangles fill the exact shape — this limit IS the definite integral.
Two meanings
1. Anti-derivative: ∫f(x)dx = F(x) + c
d/dx of F(x) = f(x)
2. Area: ∫[a→b] f(x)dx = area under curve
💡 Both meanings are linked by the Fundamental Theorem of Calculus — the area from a to b equals F(b) − F(a).
Engineering uses
  • 🚗 Distance = ∫ velocity dt
  • ⚡ Charge q = ∫ current dt
  • 🔧 Work done W = ∫ F dx
  • 🌡️ Heat energy Q = ∫ P dt
  • 📡 RMS current from AC waveform
📐

Integration Rules

Three core rules — the mirror image of differentiation rules.

The three core rules POWER RULE ∫xⁿ dx = xⁿ⁺¹/(n+1) + c Add 1 to power, divide by new power e.g. ∫x⁴ dx = x⁵/5 + c ⚠ fails for n = −1 (see ln|x|) CONSTANT MULTIPLE ∫k·f(x) dx = k·∫f(x) dx Constant factor moves outside ∫ e.g. ∫5x² dx = 5·x³/3 + c SUM RULE ∫(f+g) dx = ∫f dx + ∫g dx Integrate each term separately e.g. ∫(x²+3) dx = x³/3+3x+c

🔢 Power Rule Explorer — drag to change n and see ∫xⁿ dx update

ℹ️ How to use: Drag the slider to change the power n. Watch how the antiderivative curve changes shape as n changes — integration raises the power by 1 and divides by the new power.

2
∫x² dx = x³/3 + c

↕ Constant of Integration — drag c to see the family of curves

ℹ️ How to use: Drag the slider to change c. All the curves shift up or down together but never change shape — this is why we always write +c, the “family of curves” that share the same derivative.

0
y = x³ + 0 — all curves have the same derivative 3x²
Power rule examples
∫x6 dx = x7/7 + c
∫x3 dx = x4/4 + c
∫x dx = x2/2 + c
∫1 dx = x + c
∫x−3 dx = x−2/(−2) + c = −1/(2x2) + c
∫√x dx = ∫x½ dx = x3/2/(3/2) + c = 2x3/2/3 + c
⚠ The power rule FAILS for n = −1.
∫x−1 dx = ∫(1/x) dx = ln|x| + c
Diff vs Integration — the mirror
DIFFERENTIATE
x5 → 5x4
x3 → 3x2
x → 1
× by power, power−1
INTEGRATE
x4 → x5/5 + c
x2 → x3/3 + c
1 → x + c
power+1, ÷ by new power
Why +c?

Differentiating y = x³ + 5 or y = x³ − 7 both give 3x². So when we reverse the process we cannot know the constant — we write +c to cover all possibilities.

∫3x² dx = x³ + c (c can be any number)
If you know a point on the curve, substitute it to find c exactly.
Full polynomial
∫(4x3 − 3x2 + 2x − 5) dx
4x3 →4 · x4/4 = x4
−3x2 →−3 · x3/3 = −x3
2x →2 · x2/2 = x2
−5 →−5x
Answer:x4 − x3 + x2 − 5x + c ✓
∑

More Standard Integrals

Trig, exponential and reciprocal functions. Appear in AC circuits, oscillations and cooling problems.

∫ Standard Integrals ExplorerTrig, exponential & chain rule in reverse

ℹ️ How to use: Use the dropdown to choose a function, and type the bracket constant (a) where relevant. The solver shows the standard result and, for compound functions like sin(ax), explains the “divide by a” rule step by step.

Select a function above to see the working.
💡 The “divide by a” rule: when integrating f(ax+b), integrate as normal then divide the whole result by a — the coefficient of x inside the bracket. This undoes the chain rule.
Standard integrals to memorise TRIGONOMETRIC ∫sin x dx = −cos x + c ∫cos x dx = sin x + c ∫sec²x dx = tan x + c EXPONENTIAL ∫eˣ dx = eˣ + c ∫eᵃˣ dx = eᵃˣ/a + c divide by derivative of bracket RECIPROCAL ∫(1/x) dx = ln|x| + c ∫x⁻¹ dx = ln|x| + c special case — NOT x⁰/0
Chain rule in reverse

For integrals of the form f(ax+b): integrate normally then divide by a (the derivative of the bracket).

∫sin(3x) dx = −cos(3x)/3 + c
∫cos(2x+1) dx = sin(2x+1)/2 + c
∫e5x dx = e5x/5 + c
∫(2x+3)4 dx = (2x+3)5/10 + c
Trig worked example
∫(2 sin x + 3 cos x) dx
∫2 sin x:−2 cos x
∫3 cos x:+3 sin x
Answer:−2 cos x + 3 sin x + c ✓
∫π0 sin x dx
Integrate:[−cos x]0π
F(π):−cos π = −(−1) = 1
F(0):−cos 0 = −1
Answer:1 − (−1) = 2 ✓ (one arch of sine)
✏️

Worked Examples — Indefinite Integrals

Full step-by-step solutions. Always show the integration step before substituting any conditions.

✏️ Integration Solver — Find y from dy/dxPolynomial integration with full working

ℹ️ How to use: Type the coefficients of your polynomial dy/dx into the boxes. The solver integrates each term and shows every step. Optionally enter a point (x, y) the curve passes through to find the exact value of c.

Enter the coefficients above.
👆 Try: 6x²+4x−3 (indefinite) · dy/dx=6x²+2x, passes (1,5)
📌 For indefinite integrals always include + c. If a point is given, substitute it to find c.
Example 1 — Basic polynomial: ∫(6x2 + 4x − 3) dx
6x2 →6 · x3/3 = 2x3
4x →4 · x2/2 = 2x2
−3 →−3x
Answer:2x3 + 2x2 − 3x + c ✓
Example 2 — Negative power: ∫(3/x2) dx = ∫3x−2 dx
Rewrite:3 · x−2
Integrate:3 · x−1/(−1) = −3x−1
Answer:−3/x + c ✓
Example 3 — Find c from a condition: dy/dx = 6x + 2, curve passes through (1, 5)
Integrate:y = 3x2 + 2x + c
Substitute (1,5):5 = 3(1) + 2(1) + c = 5 + c
So c =0
Answer:y = 3x2 + 2x ✓
Example 4 — Double integration: d2y/dx2 = 12x − 4.  dy/dx = 8 at x = 1.  y = 3 at x = 0.
1st integral:dy/dx = 6x2 − 4x + A
Use dy/dx(1) = 8:8 = 6 − 4 + A → A = 6
2nd integral:y = 2x3 − 2x2 + 6x + B
Use y(0) = 3:B = 3
Answer:y = 2x3 − 2x2 + 6x + 3 ✓
🎮

Integration Games

Two games to drill the power rule and standard integrals.

⚡ POWER RULE SPEED ROUND — 10 Questions

ℹ️ How to play: An integration question appears. Click the correct answer. Score and streak update after each — click Next to continue.

Score: 0 Streak: 0 🔥 Best: 0 Q 1/10
Choose the correct integral:
∫x³ dx

🔗 MATCH UP — Function to Standard Integral

Tap a function on the left, then its integral on the right

⚙️

Applications

Where this topic is used in engineering, manufacturing, maintenance and daily life.

⚙️ Engineering Integration CalculatorCharge, work, velocity, displacement

ℹ️ How to use: Pick a real engineering scenario from the dropdown. Type the time/distance limits and the solver integrates the function and evaluates the definite integral — showing every step.

Select a scenario above to see the working.
👆 Try: Charge from t=0 to 3s · Work from x=0 to 4m · Velocity from t=0 to 5s
📝

Practice Questions

Show all working. Always include + c for indefinite integrals.

🟢 Tier 1 — Basic

ℹ️ How to use: Work through each question and write down your answer. When ready, click Show Answer for the full step-by-step solution.

Q1
Find: (a) ∫x5 dx   (b) ∫6x2 dx   (c) ∫(5x3 − 4x + 7) dx   (d) ∫x−4 dx
[6]
(a) x6/6 + c
(b) 2x3 + c
(c) 5x4/4 − 2x2 + 7x + c
(d) −x−3/3 + c = −1/(3x3) + c
Q2
Integrate: (a) ∫√x dx   (b) ∫(1/x3) dx   (c) ∫(3√x − 2/x2) dx
[5]
(a) ∫x½ dx = 2x3/2/3 + c
(b) ∫x−3 dx = −x−2/2 + c = −1/(2x2) + c
(c) 3·2x3/2/3 − 2·(−x−1) + c = 2x3/2 + 2/x + c
🟡 Tier 2 — Finding c
Q3
The gradient of a curve is dy/dx = 3x2 − 6x. The curve passes through (2, 1). Find y.
[4]
y = x3 − 3x2 + c
At (2,1): 1 = 8 − 12 + c → c = 5
y = x3 − 3x2 + 5 ✓
Q4
d2y/dx2 = 6x. Given dy/dx = 4 when x = 1, and y = 0 when x = 0. Find y.
[5]
1st ∫: dy/dx = 3x2 + A. At x=1: 4 = 3 + A → A = 1
2nd ∫: y = x3 + x + B. At x=0: 0 = B → B = 0
y = x3 + x ✓
🔴 Tier 3 — Standard functions
Q5
Integrate: (a) ∫sin(4x) dx   (b) ∫e3x dx   (c) ∫(2cos x − 3/x) dx
[6]
(a) −cos(4x)/4 + c
(b) e3x/3 + c
(c) 2 sin x − 3 ln|x| + c
🎯 Score guide: Q1–2 foundation (11 marks) · Q3–4 applied (9 marks) · Q5 challenge (6 marks)
⏱ Now test yourself — 15 seconds per question!
SL

SkillLondon — T Level / Level 3 Engineering Maths  ·  Integration 1 — Rules, Standard Integrals & Worked Examples