Integration (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
Test yourself on Integration (AS Unit 1: Pure Mathematics A) with WJEC A-Level practice questions.
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Integration (AS Unit 1: Pure Mathematics A) explained
The power rule of integration reverses the power rule of differentiation: for any rational power n ≠ −1, ∫ xⁿ dx = (x^(n+1)) / (n + 1) + c, where c is the constant of integration.
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The case n = −1 is excluded because dividing by n + 1 = 0 is undefined (its antiderivative is ln|x|). Integration is linear: constants factor out, ∫ k f(x) dx = k ∫ f(x) dx, and sums and differences integrate term by term, ∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x) dx. Before integrating, algebraic expressions must be simplified into individual power terms Axⁿ by expanding products such as (2x + 1)(x − 3) into 2x² − 5x − 3, and splitting fractions over a single monomial denominator such as (3x⁴ + 1) / x² = 3x² + x⁻². The constant of integration + c must never be omitted.
Your focus
- Integrate powers of x for any rational index n ≠ −1 using the power rule.
- Simplify products and algebraic fractions into separate power terms prior to integration.
- Include and manage the arbitrary constant of integration + c correctly.
Integration (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- preparing the integrand by expanding brackets and splitting monomial denominators into powers of x
- applying the power rule ∫ xⁿ dx = (x^(n+1))/(n + 1) to each term
- including the arbitrary constant of integration + c in the final expression
Examiner Tips
- 💡Rewrite all surds as fractional exponents and all reciprocals as negative exponents before integrating.
- 💡Check your integration by differentiating your final answer to see if you recover the original integrand.
Common Mistakes
- integrating each factor in a product separately instead of expanding brackets first
- making arithmetic errors when adding 1 to negative or fractional exponents, such as −2 + 1 = −3