Why integration matters in everyday maths
Ever wondered how you can find the area under a curve without sketching it? That’s what integration does – it adds up infinitely tiny pieces, just like counting raindrops to know the total rain.
💡 In Simple Words: Integration is the reverse of differentiation. It tells you a function whose slope matches the one you’re given. Think of it as rebuilding a hill when you only know how steep each part is.
Power Rule for Integration
The power rule lets you integrate any term that looks like xⁿ (x raised to a power). The rule says: increase the exponent by one, then divide by the new exponent. Write it as:
∫xⁿdx = xⁿ⁺¹/(n+1) + C, where C is the constant of integration (the “+ C” you always add because many functions share the same derivative).
Example: Find ∫3x²dx.
- Identify the power: n = 2.
- Raise the exponent: 2 + 1 = 3.
- Divide by the new exponent: 3x³/3.
- Don’t forget + C.
Result: ∫3x²dx = x³ + C.
Constant Multiple Rule
If a constant sits in front of a function, you can pull it out of the integral. It’s like taking a weight off a cart before you start pushing.
Rule: ∫k·f(x)dx = k·∫f(x)dx, where k is any number.
Example: ∫5·sin x dx.
- Pull out the 5: 5·∫sin x dx.
- Integral of sin x is –cos x.
- Put it back: 5·(–cos x) + C.
Result: ∫5sin x dx = –5cos x + C.
Sum/Difference Rule
When you have a sum or difference of functions, integrate each part separately. It’s like solving a puzzle piece by piece.
Rule: ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx.
Example: ∫(2x³ – 4x)dx.
- Split: ∫2x³dx – ∫4x dx.
- Apply power rule to each: (2·x⁴/4) – (4·x²/2).
- Simplify: (½)x⁴ – 2x² + C.
Result: ∫(2x³ – 4x)dx = ½x⁴ – 2x² + C.
Exponential Rule
Exponential functions look like aⁿ where a is a constant (e.g., eˣ or 2ˣ). Their integral is almost the same shape, just divided by a constant.
Rule for eˣ (the natural exponential): ∫eˣdx = eˣ + C. For aⁿ where a ≠ e: ∫aˣdx = aˣ/ln a + C, where ln a is the natural logarithm of a.
Example: ∫3·2ˣdx.
- Pull out 3: 3·∫2ˣdx.
- Integral of 2ˣ is 2ˣ/ln 2.
- Multiply back: 3·2ˣ/ln 2 + C.
Result: ∫3·2ˣdx = (3·2ˣ)/ln 2 + C.
Trigonometric Rule
Trig functions (like sin x, cos x, tan x) have neat integrals that mirror each other.
- ∫sin x dx = –cos x + C
- ∫cos x dx = sin x + C
- ∫sec²x dx = tan x + C
- ∫csc²x dx = –cot x + C
Example: ∫(4cos x + 7)dx.
- Separate: 4∫cos x dx + 7∫dx.
- Integrate: 4·sin x + 7·x.
- Add + C.
Result: ∫(4cos x + 7)dx = 4sin x + 7x + C.
Quick Comparison of Basic Rules
| Rule | Form of f(x) | Result (∫f(x)dx) |
|---|---|---|
| Power | xⁿ (n≠‑1) | xⁿ⁺¹/(n+1) + C |
| Constant Multiple | k·f(x) | k·∫f(x)dx + C |
| Sum/Difference | f(x) ± g(x) | ∫f(x)dx ± ∫g(x)dx + C |
| Exponential (eˣ) | eˣ | eˣ + C |
| Exponential (aˣ) | aˣ (a>0, a≠1) | aˣ/ln a + C |
| Trig – sin | sin x | –cos x + C |
| Trig – cos | cos x | sin x + C |
How to pick the right rule – a simple flow
📝 Likely Exam Questions
- Find ∫(5x⁴ – 2x⁻³)dx.
Answer: (5·x⁵/5) – (2·x⁻²/‑2) + C = x⁵ + x⁻² + C. - Evaluate ∫7eˣdx.
Answer: 7eˣ + C. - Compute ∫(3sin x + 4cos x)dx.
Answer: –3cos x + 4sin x + C. - Integrate ∫2·3ˣdx.
Answer: 2·3ˣ/ln 3 + C. - Using the sum rule, integrate ∫(x² + 2x + 1)dx.
Answer: x³/3 + x² + x + C.