Indefinite & Definite Integrals
CSEC Additional Mathematics Essential Knowledge: Integration is the reverse process of differentiation. It allows us to find functions when given their derivatives, calculate areas under curves, and solve problems involving accumulation. Mastering integrals is crucial for CSEC Add Maths and lays the foundation for higher mathematics.
Key Concept: Integration has two main types: Indefinite Integrals (finding antiderivatives + constant) and Definite Integrals (calculating numerical values representing areas). The fundamental theorem of calculus connects these two concepts.
Part 1: Indefinite Integrals – The Antiderivative
Basic Integration Rules
If \(F'(x) = f(x)\), then \(\int f(x) \, dx = F(x) + C\), where \(C\) is the constant of integration.
The integral sign ∫, the integrand \(f(x)\), and \(dx\) (indicating integration with respect to \(x\)) are essential components.
| Function | Integral | Rule Name |
|---|---|---|
| \(\int x^n \, dx\) | \(\frac{x^{n+1}}{n+1} + C, \quad n \neq -1\) | Power Rule |
| \(\int k \, dx\) | \(kx + C\) | Constant Rule |
| \(\int e^x \, dx\) | \(e^x + C\) | Exponential Rule |
| \(\int \frac{1}{x} \, dx\) | \(\ln|x| + C\) | Logarithmic Rule |
| \(\int \sin x \, dx\) | \(-\cos x + C\) | Trigonometric Rules |
| \(\int \cos x \, dx\) | \(\sin x + C\) | Trigonometric Rules |
Remember: Always add \(+C\) for indefinite integrals! This constant represents all possible vertical shifts of the antiderivative.
Find \(\int (3x^2 + 4x – 5) \, dx\)
Find \(\int \left( \sqrt{x} + \frac{1}{x^2} \right) dx\)
Part 2: Definite Integrals – Area Under Curves
The Definite Integral Concept
The definite integral \(\int_a^b f(x) \, dx\) represents the net signed area between the curve \(y = f(x)\), the x-axis, and the vertical lines \(x = a\) and \(x = b\).
where \(F(x)\) is any antiderivative of \(f(x)\). This is the Fundamental Theorem of Calculus.
| Property | Formula | Meaning |
|---|---|---|
| Reversal of Limits | \(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\) | Swapping limits changes sign |
| Zero Width | \(\int_a^a f(x) \, dx = 0\) | No area under a point |
| Additivity | \(\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx\) | Can split integration interval |
| Constant Multiple | \(\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx\) | Constants can be factored out |
| Sum/Difference | \(\int_a^b [f(x) \pm g(x)] \, dx = \int_a^b f(x) \, dx \pm \int_a^b g(x) \, dx\) | Integrate term by term |
Evaluate \(\int_1^3 (2x + 1) \, dx\)
Evaluate \(\int_0^2 3x(1 + x^2)^2 \, dx\)
Part 3: Applications – Area Under/Between Curves
Calculating Areas with Integration
To find the area between \(y = f(x)\) and the x-axis from \(x = a\) to \(x = b\):
Important: If the curve goes below the x-axis, the definite integral gives net signed area (positive above, negative below). For total area, take the absolute value or integrate sections separately.
If \(f(x) \geq g(x)\) on \([a, b]\), the area between them is:
Find the area bounded by the curve \(y = x^2 – 4x + 3\), the x-axis, and the lines \(x = 0\) and \(x = 3\).
Find the area enclosed by the curves \(y = x^2\) and \(y = 2x – x^2\).
Part 4: Integration Techniques for CSEC
Essential Integration Methods
1. Term-by-Term Integration
For polynomials and sums: Integrate each term separately using the power rule.
Example: \(\int (3x^2 + 2x + 1) dx = x^3 + x^2 + x + C\)
CSEC Tip: Rewrite roots as fractional powers and reciprocals as negative powers first.
2. Substitution Method
For composite functions: Let \(u = g(x)\), then \(du = g'(x) dx\).
Example: \(\int 2x e^{x^2} dx\): Let \(u = x^2\), \(du = 2x dx\) → \(\int e^u du = e^u + C = e^{x^2} + C\)
CSEC Level: Usually simple substitutions like \(u = ax + b\) or \(u = x^2\).
3. Integration of Trigonometric Functions
Memorize basic integrals: \(\int \sin x dx = -\cos x + C\), \(\int \cos x dx = \sin x + C\)
CSEC Focus: Usually combined with substitution, e.g., \(\int \sin(2x) dx = -\frac{1}{2}\cos(2x) + C\)
| Function | Integral | Notes |
|---|---|---|
| \(\int (ax + b)^n dx\) | \(\frac{(ax + b)^{n+1}}{a(n+1)} + C, \quad n \neq -1\) | Use substitution or formula |
| \(\int e^{kx} dx\) | \(\frac{1}{k} e^{kx} + C\) | |
| \(\int \frac{1}{ax + b} dx\) | \(\frac{1}{a} \ln|ax + b| + C\) | |
| \(\int \sin(ax + b) dx\) | \(-\frac{1}{a} \cos(ax + b) + C\) | |
| \(\int \cos(ax + b) dx\) | \(\frac{1}{a} \sin(ax + b) + C\) |
Find \(\int (2x + 1)^3 dx\)
Part 5: Solving Differential Equations (CSEC Level)
Basic Differential Equations
The simplest differential equation: Integrate both sides with respect to \(x\).
If given an initial condition (e.g., \(y = y_0\) when \(x = x_0\)), use it to find the constant \(C\).
Solve \(\frac{dy}{dx} = 3x^2 – 2x\), given that \(y = 4\) when \(x = 1\).
The rate of change of the area \(A\) of a culture of bacteria is given by \(\frac{dA}{dt} = 3t^2 + 2t\), where \(t\) is time in hours. If \(A = 5\) when \(t = 1\), find \(A\) in terms of \(t\).
Quiz: Test Your Integration Skills
Term-by-term integration:
\(\int 4x^3 dx = x^4\)
\(\int -3x^2 dx = -x^3\)
\(\int 2x dx = x^2\)
\(\int -1 dx = -x\)
So: \(x^4 – x^3 + x^2 – x + C\)
\(\int \cos x dx = \sin x + C\)
So \(\int_0^{\pi/2} \cos x dx = \left[ \sin x \right]_0^{\pi/2} = \sin(\pi/2) – \sin(0) = 1 – 0 = 1\)
Since \(y = x^3 > 0\) on [1, 2], area = \(\int_1^2 x^3 dx = \left[ \frac{x^4}{4} \right]_1^2\)
= \(\frac{16}{4} – \frac{1}{4} = 4 – 0.25 = 3.75\) or \(\frac{15}{4}\) square units.
\(y = \int \frac{1}{x} dx = \ln|x| + C\)
Using condition: When \(x = e\), \(y = 2\): \(2 = \ln e + C = 1 + C \Rightarrow C = 1\)
So \(y = \ln|x| + 1\) (or \(\ln x + 1\) for \(x > 0\))
First expand: \((2x + 1)(x – 3) = 2x^2 – 6x + x – 3 = 2x^2 – 5x – 3\)
Then integrate: \(\frac{2x^3}{3} – \frac{5x^2}{2} – 3x + C\)
🎯 Key Concepts Summary
- Indefinite Integral: \(\int f(x) dx = F(x) + C\) where \(F'(x) = f(x)\)
- Definite Integral: \(\int_a^b f(x) dx = F(b) – F(a)\) = net signed area
- Power Rule: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)
- Common Integrals:
- \(\int e^x dx = e^x + C\)
- \(\int \frac{1}{x} dx = \ln|x| + C\)
- \(\int \sin x dx = -\cos x + C\)
- \(\int \cos x dx = \sin x + C\)
- Area Calculations:
- Under curve: \(\int_a^b |f(x)| dx\)
- Between curves: \(\int_a^b [f(x) – g(x)] dx\) where \(f(x) \geq g(x)\)
- Solving \(\frac{dy}{dx} = f(x)\): Integrate and use initial condition to find \(C\)
- CSEC Exam Tips:
- Always add \(+C\) for indefinite integrals
- For definite integrals, show substitution of limits clearly
- Rewrite functions in power form before integrating
- For area problems, sketch if possible to identify regions
- Check your answer by differentiating (when time permits)
CSEC Exam Strategy: Integration questions often appear in Paper 2 (structured questions). Common question types: (1) Find indefinite integrals, (2) Evaluate definite integrals, (3) Calculate areas under/between curves, (4) Solve simple differential equations. Always show your working: write the integral, show the antiderivative, substitute limits (for definite integrals), and simplify. For area questions, if the curve crosses the x-axis, split the integral at the intersection points.