Integration is the reverse of differentiation. This guide covers all the integration techniques you need for O Level For the best math tuition Singapore, explore our structured programs designed for exam success. A Math.
What Is Integration?
Integration finds the area under a curve. It’s the reverse of differentiation.
If dy/dx = f(x), then y = ∫f(x)dx + C
The C is the constant of integration – always include it for indefinite integrals.
Basic Integration Rules
- Power Rule
- ∫x^n dx = x^(n+1)/(n+1) + C (for n ≠ -1)
Examples:
- ∫x^3 dx = x^4/4 + C
- – ∫x^5 dx = x^6/6 + C
- – ∫x dx = x^2/2 + C
2. Constant Multiple Rule
∫kf(x) dx = k∫f(x) dx
Examples:
- ∫5x^2 dx = 5 × x^3/3 + C = 5x^3/3 + C
- – ∫3x^4 dx = 3x^5/5 + C
3. Sum/Difference Rule
Integrate each term separately.
Examples:
- ∫(x^3 + 2x) dx = x^4/4 + x^2 + C
- – ∫(4x^2 – x) dx = 4x^3/3 – x^2/2 + C
4. Constant Rule
∫k dx = kx + C
Examples:
- ∫5 dx = 5x + C
- – ∫7 dx = 7x + C
Special Cases
Negative Powers:
∫x^(-n) dx = x^(-n+1)/(-n+1) + C
Example: ∫x^(-2) dx = x^(-1)/(-1) + C = -1/x + C
Fractional Powers:
Same rule applies.
Example: ∫x^(1/2) dx = x^(3/2)/(3/2) + C = 2x^(3/2)/3 + C
Definite Integration
Definite integrals have limits and give a numerical value.
∫[a to b] f(x) dx = F(b) – F(a)
Where F(x) is the antiderivative.
Example: ∫[1 to 3] x^2 dx
= [x^3/3] from 1 to 3
= (27/3) – (1/3)
= 9 – 1/3
= 26/3
Finding Constants
Given dy/dx and a point, find C.
Example: dy/dx = 2x + 1, and y = 5 when x = 1
y = ∫(2x + 1) dx = x^2 + x + C
Substitute x = 1, y = 5:
5 = 1 + 1 + C
C = 3
So y = x^2 + x + 3
Integration of Special Functions
∫1/x dx = ln|x| + C
∫e^x dx = e^x + C
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
Note: These may or may not be in your syllabus. Check your syllabus requirements.
Area Under a Curve
The area under y = f(x) from x = a to x = b is:
Area = ∫[a to b] y dx
If the curve is below the x-axis, the integral is negative. Take the absolute value for area.
Area Between Two Curves
Area between y = f(x) and y = g(x) from x = a to x = b:
Area = ∫[a to b] [f(x) – g(x)] dx
Where f(x) is the upper curve and g(x) is the lower curve.
Kinematics Applications
Given velocity v = f(t):
- Displacement s = ∫v dt
- – Distance travelled = ∫|v| dt
Given acceleration a = f(t):
- Velocity v = ∫a dt
Common Mistakes
Forgetting + C:
Always include the constant of integration for indefinite integrals.
Wrong limits order:
∫[a to b] ≠ ∫[b to a]
Calculation errors:
Use brackets carefully when substituting limits.
Not checking units:
In kinematics, check if the answer should be in seconds, metres, etc.
Practice Tips
Master basic integration first.
Practise finding constants.
Work on area problems.
Connect to kinematics applications.
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