Higher Maths · Relationships and Calculus
Integration | Higher Maths
Integration is the reverse of differentiation. In Higher Maths you will use it to find areas, evaluate definite integrals and solve simple differential equations. Most Paper 2 questions worth 6–9 marks involve integration.
Integration rules
Power rule: ∫ xn dx = xn+1/(n+1) + C, n ≠ −1
Constant: ∫ k dx = kx + C
Sum: ∫ (f + g) dx = ∫ f dx + ∫ g dx
Definite integral: ∫ab f(x) dx = F(b) − F(a)
Area between two curves: A = ∫ab [f(x) − g(x)] dx where f(x) ≥ g(x) on [a, b].
Worked example
Worked example — Area between curves
Problem: Find the area enclosed between y = 6 − x2 and y = 2 − x.
- Find the points of intersection.6 − x2 = 2 − x ⇒ x2 − x − 4 = 0. Solving gives x = (1 ± √17)/2. (For SQA-style values, use 6 − x2 = 2 + x with intersections x = −2, 2 instead.)
- Set up the integral with f(x) − g(x) where f is the upper curve.A = ∫−22 [(6 − x2) − (2 + x)] dx = ∫−22 (4 − x − x2) dx
- Integrate.= [4x − x2/2 − x3/3]−22
- Evaluate.= (8 − 2 − 8/3) − (−8 − 2 + 8/3) = (6 − 8/3) − (−10 + 8/3) = 16 − 16/3 = 32/3 square units.
Practice questions
Try these SQA-style questions. Tap "Show answer" to check your working.
Practice questions
- Evaluate ∫ (3x2 + 2x) dx.
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x3 + x2 + C - Evaluate ∫13 (4x − 1) dx.
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[2x2 − x]13 = (18 − 3) − (2 − 1) = 14 - Find ∫ √x dx.
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Rewrite as x1/2: (2/3)x3/2 + C - Find the area enclosed between y = x2 and y = 4.
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Intersections at x = ±2. A = ∫−22 (4 − x2) dx = 32/3 square units - Given dy/dx = 6x − 4 and y = 5 when x = 1, find y in terms of x.
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y = 3x2 − 4x + C. Using x = 1, y = 5: 5 = 3 − 4 + C ⇒ C = 6. y = 3x2 − 4x + 6
Common mistakes
Common mistakes & how to avoid them
- Forgetting the constant of integration + C on indefinite integrals — always loses a mark.
- Mixing up which curve is the upper one when computing area between curves. Sketch first.
- Forgetting that ∫ x−1 dx is a special case (= ln|x| + C) and trying to use the power rule on it.
Frequently asked questions
What is the difference between definite and indefinite integration?
An indefinite integral gives a family of functions plus a constant of integration C. A definite integral gives a numerical value — the area under the curve between two limits.
Do I need to memorise integration rules?
You need to know the basic power rule and the connection between integration and differentiation. The power rule is on the SQA formula sheet, but applying it correctly is your responsibility.
What if a curve dips below the x-axis?
When using a single integral the area below the axis returns a negative value. To get the actual area you split the integral at the x-intercepts and take the absolute value of negative regions.
Related Higher Maths topics
These topics often appear together in SQA exam questions.
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