Integration as the reverse of differentiation, indefinite integrals and the constant of integration, the rules, integrating powers, (ax + b)ⁿ and trigonometric functions.
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Differentiate and you lose the constant. Integrate and you must put it back. Integration is just differentiation run backwards — and checking is always possible.
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Copy these into your notebook in your own words — that is how they stick.
If F′(x) = f(x) then ∫ f(x) dx = F(x) + C.
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1.
∫ cf dx = c∫f dx; ∫(f ± g) dx = ∫f dx ± ∫g dx.
∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹/[a(n + 1)] + C.
∫ sin px dx = −cos px/p + C; ∫ cos qx dx = sin qx/q + C.
C represents a family of functions differing by constants.
Roots and reciprocals as powers; expand products.
Differentiate the answer.
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8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (24 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [13 marks]
Find each of the following, where C is an arbitrary constant.
(a) ∫ (4x³ − 6x + 5) dx [2]
(b) ∫ (√x + 3/x²) dx [3]
(c) ∫ (2x + 1)³ dx [3]
(d) ∫ (3 sin 2x − 4 cos x) dx [3]
(e) Show, by differentiation, that your answer to part (c) is correct. [2]
Question 2 [11 marks]
Find each integral.
(a) ∫ (5 − 2x)⁴ dx [3]
(b) ∫ (sin 4x + cos(x/2)) dx [3]
(c) ∫ x(x² + 1) dx [3]
(d) Explain why the answer to an indefinite integral must include a constant C. [2]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.