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This lesson covers indefinite integration — the reverse process of differentiation — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to be able to integrate standard functions, apply the power rule for integration, understand the constant of integration, and recognise standard results.
Differentiation takes a function and produces its derivative (the rate of change). Integration is the reverse process — it takes a derivative and recovers the original function (up to a constant).
If dy/dx = f(x), then y = ∫ f(x) dx
The symbol ∫ is the integral sign, f(x) is the integrand, and dx tells us we are integrating with respect to x.
When we differentiate y = x³, we get dy/dx = 3x². So if we are told that dy/dx = 3x², we can work backwards and say y = x³ + c, where c is a constant. This "working backwards" is integration.
Key Point: Integration and differentiation are inverse operations. If you differentiate a function and then integrate the result, you get back to the original function (plus a constant).
The most fundamental integration rule is the power rule. It reverses the power rule for differentiation.
If dy/dx = xⁿ (where n ≠ -1), then:
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + c
In words: increase the power by 1, then divide by the new power, and add a constant of integration.
Example 1: Find ∫ x⁴ dx
Example 2: Find ∫ x⁻³ dx
Example 3: Find ∫ √x dx
Example 4: Find ∫ 1/x² dx
Exam Tip: Always rewrite roots and fractions as powers of x before integrating. For example, √x = x^(1/2), 1/x³ = x⁻³, and ³√x = x^(1/3).
When we differentiate y = x³ + 7, we get dy/dx = 3x². But we also get dy/dx = 3x² when we differentiate y = x³ + 100, or y = x³ - 42, or y = x³ + any constant.
This means that when we integrate 3x², we cannot determine the original constant. We write:
∫ 3x² dx = x³ + c
The constant of integration c represents all possible constant values. It is essential to include c in every indefinite integral.
If we are given a boundary condition (a known point on the curve), we can find the specific value of c.
Example: Given dy/dx = 4x - 3 and the curve passes through (2, 5), find y.
Exam Tip: Forgetting the constant of integration is one of the most common mistakes. Always write + c for indefinite integrals. Only omit c when evaluating definite integrals (the constants cancel).
∫ k × f(x) dx = k × ∫ f(x) dx
You can take a constant factor outside the integral.
Example: ∫ 5x³ dx = 5 × ∫ x³ dx = 5 × (x⁴/4) + c = 5x⁴/4 + c
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
You can integrate term by term.
Example: ∫ (3x² + 2x - 7) dx = x³ + x² - 7x + c
The following standard integrals appear frequently at A-Level:
| Function f(x) | Integral ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ -1) | xⁿ⁺¹ / (n + 1) + c |
| 1/x (i.e. x⁻¹) | ln |
| eˣ | eˣ + c |
| e^(kx) | (1/k)e^(kx) + c |
| sin x | -cos x + c |
| cos x | sin x + c |
| sec²x | tan x + c |
| sin(kx) | -(1/k)cos(kx) + c |
| cos(kx) | (1/k)sin(kx) + c |
Example 1: ∫ 1/x dx = ln|x| + c
Note the modulus sign — we need |x| because ln is only defined for positive values.
Example 2: ∫ e^(3x) dx = (1/3)e^(3x) + c
Example 3: ∫ cos(2x) dx = (1/2)sin(2x) + c
Example 4: ∫ (3eˣ + 2/x - sin x) dx = 3eˣ + 2 ln|x| + cos x + c
Sometimes you need to expand or simplify before integrating.
Example 1: Find ∫ x(x + 3) dx
Example 2: Find ∫ (2x + 1)² dx (when not using substitution)
Example 3: Find ∫ (x² + 3)/x dx
Exam Tip: You cannot integrate a product or a quotient term by term (i.e., ∫ f(x)g(x) dx ≠ ∫ f(x) dx × ∫ g(x) dx). You must expand or simplify first, or use a technique like substitution or parts.
A common exam question gives you a gradient function and a point on the curve, then asks for the equation.
Example: A curve has gradient function dy/dx = 6x² - 4x + 1 and passes through the point (1, 3). Find the equation of the curve.