A-level integration questions, from first steps to exam standard

What you are given, and what you must know

AQA's formula booklet gives integration by parts, the integral of f′(x)/f(x), and the integrals of tan x and cot x. Edexcel's specification lists the integrals you must memorise: xⁿ, sin kx, cos kx, eᵏˣ and 1/x, plus the reverse of the chain rule (Appendix 1). Learn those by heart.

Level 1: the power rule

Question 1: find ∫ (6x² − 4x + 3) dx.

Raise each power by one and divide by the new power: 6x³ ÷ 3 − 4x² ÷ 2 + 3x, which is 2x³ − 2x² + 3x + c. The + c is worth a mark on an indefinite integral.

Level 2: definite integrals and area

Question 2: the curve y = 9 − x² and the two coordinate axes enclose a region in the first quadrant. Find its area.

The area is 18 square units. In the graph, x runs across and y up: the curve meets the x-axis at 3, so the whole region lies above it and the integral is the area. Where part of a region dips below the axis, that part integrates to a negative number, so work out each piece on its own and add their sizes.

Level 3: exponentials and trigonometry

Question 3: find ∫ (e³ˣ + cos 2x) dx.

Divide by the coefficient of x each time: ⅓e³ˣ + ½ sin 2x + c. Check by differentiating, which multiplies the 3 and the 2 back in. Trigonometric integrals only work in radians.

Level 4: spotting f′(x) over f(x)

Question 4: find ∫ 2x ÷ (x² + 5) dx. The top is exactly the derivative of the bottom, so the answer is ln(x² + 5) + c. When the top is a multiple of the derivative, adjust: ∫ x ÷ (x² + 5) dx = ½ ln(x² + 5) + c. Our reverse chain rule guide covers this pattern in depth.

Level 5: integration by parts

Question 5: find ∫ x cos x dx. Let u = x, so du/dx = 1, and dv/dx = cos x, so v = sin x.

Choose u as the part that gets simpler when differentiated. The one exception is ln x, which you take as u because you can't easily integrate it. More examples are in our integration by parts guide.

Level 6: integration by substitution

Question 6: use u = x² + 1 to find ∫₀¹ x(x² + 1)³ dx.

1. du/dx = 2x, so x dx = ½ du.
2. Change the limits: x = 0 gives u = 1, and x = 1 gives u = 2.
3. The integral becomes ½∫₁² u³ du = ½[u⁴ ÷ 4] from 1 to 2.
4. That is ½(16/4 − 1/4) = 15/8.

Changing the limits saves you from substituting back. Forgetting to change them is the most common slip in substitution questions. Check any of these in the Integral Calculator.

Common integration mistakes

• Leaving out + c on an indefinite integral.
• Multiplying instead of dividing by the coefficient: ∫ e³ˣ dx is ⅓e³ˣ, not 3e³ˣ.
• Sign errors with sin and cos. ∫ sin x dx = −cos x + c.
• Using the power rule on 1/x. n = −1 is the exception: the answer is ln|x|.
• Degrees in calculus. Set your calculator to radians.
• Keeping the old limits after a substitution.

Use a daily paper as a short timed mock

For more integration questions, sit today's A-level Maths: Calculus paper, which mixes differentiation and integration. For dates, papers and the full formula booklet, see the A-level Maths guide, and to grade a full mock, the A-level Maths mocks guide.

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