Integration — The Reverse of Differentiation
Learn integration from scratch — the reverse of differentiation. Understand indefinite and definite integrals, the +C constant, and finding areas under curves.
If differentiation tells you the gradient, integration goes backwards— it undoes differentiation. It's also used to find the area under a curve.
The Power Rule (Reversed)
In English: add 1 to the power, then divide by the new power. Always add + C.
| Function | Integral |
|---|---|
| x³ | x⁴/4 + C |
| 2x | x² + C |
| 5 | 5x + C |
| 6x² | 2x³ + C |
Why + C?
When you differentiate, constants disappear (e.g. x² + 3 and x² + 99 both give 2x). So when you reverse it, you don't know what the constant was. +C is a placeholder for that unknown constant. This is called an indefinite integral.
Worked Example 1 — Indefinite Integral
Find ∫ (3x² + 4x − 1) dx
- 3x² → 3 × x³/3 = x³
- 4x → 4 × x²/2 = 2x²
- −1 → −x
- Answer: x³ + 2x² − x + C
Definite Integrals (Finding Areas)
A definite integral has limits (numbers at the top and bottom of the ∫ sign). It gives you the exact area under the curve between two x-values.
Integrate, then substitute the top limit, subtract the bottom limit. No + C needed.
Worked Example 2 — Definite Integral
Find ∫₁³ 2x dx
- Integrate: 2x → x²
- Upper limit: 3² = 9
- Lower limit: 1² = 1
- Area = 9 − 1 = 8 square units
Worked Example 3 — Finding Area Under a Curve
Find the area under y = x² + 1 between x = 0 and x = 3.
- ∫₀³ (x² + 1) dx
- Integrate: x³/3 + x
- At x = 3: 27/3 + 3 = 9 + 3 = 12
- At x = 0: 0 + 0 = 0
- Area = 12 − 0 = 12 square units
Real-World Connection
If you have a speed-time graph, the area under the curve gives you the total distance travelled. Integration turns a rate of change back into a total.
Try It Yourself
- Find ∫ (6x² − 4x + 3) dx. (Answer: 2x³ − 2x² + 3x + C)
- Find ∫₂⁴ 3x² dx. (Answer: 64 − 8 = 56)
- Find the area under y = 4x between x = 0 and x = 5. (Answer: 50)