About this chapter
Integration closes Pure 1, and on the WMA11 paper it rarely appears on its own. A typical question gives f′(x) or even f′′(x), adds a tangent, a normal or a point on the curve, and expects you to turn each piece of information into an equation for an unknown constant.
This chapter covers indefinite integration of any power of x except minus one, rewriting fractions and brackets before integrating, the constant of integration, and finding f(x) from f′(x). Definite integrals and areas are in Pure 2, so they are not used. The later questions find a constant from a normal, solve two gradient conditions simultaneously, integrate twice from f′′(x), and show that a tangent meets the curve only once.
The ten questions
- Q1 (4 marks): integrate 6x to the power 5, plus 4/x³, minus 3/√x, with each term simplified.
- Q2 (5 marks): split (3x to the power 4, minus 2) over x²√x into powers of x, then integrate.
- Q3 (5 marks): show that (2√x − 3)²/√x has the form 4√x + A + B/√x, then integrate it.
- Q4 (5 marks): find f(x) from f′(x) = 3√x − 4/x² and the point (4, 10).
- Q5 (5 marks): differentiate and then integrate the same expression, 2x³ − 6/√x.
- Q6 (9 marks): find a cubic from its gradient and a point, show it equals (x−1)²(x−2), then sketch it.
- Q7 (8 marks): a normal to the curve fixes k in f′(x), then integrate to find f(x).
- Q8 (8 marks): a tangent and a normal give two equations in a and b, then f(x).
- Q9 (8 marks): from f′′(x) and a tangent line: the normal, then integrate twice with a constant each time.
- Q10 (10 marks): find f(x) from a gradient and a point, the tangent at x = 1, and show by algebra that it meets the curve nowhere else.
Key skills tested
Integrating powers of x. Raise the power by one, then divide by the new power, for any power except minus one.
Rewrite first. Split fractions and expand brackets before integrating; never integrate the top and bottom separately. For example, (x³ + 2)/x² becomes x + 2/x².
Negative and fractional powers. Dividing by a fraction means multiplying by its reciprocal. For example, 1/√x integrates to 2√x + c.
The constant c. Every indefinite integral needs + c, and every coefficient should be simplified.
f(x) from f′(x). Integrate, then substitute the given point to find c, and write f(x) out in full.
Integrating twice. From f′′(x), integrate once and use a gradient to find the first constant, then integrate again and use a point for the second.
Using tangents and normals. A tangent at x = a gives both f(a) and f′(a). A normal with gradient m tells you f′(a) = −1/m.
Unknown constants. Each condition gives one equation, so two unknowns need two conditions, solved simultaneously.
Check by differentiating. Differentiate your answer; you should get back the expression you integrated.
Worked example
Question 4 from this chapter. The curve y = f(x), x > 0, has f′(x) = 3√x − 4/x² and passes through (4, 10). Find f(x).
Write each term as a power of x: 3√x is 3 times x to the power ½, and 4/x² is 4 times x to the power −2.
Integrate each term. Raising ½ by one gives 3/2, and 3 ÷ 3/2 = 2, so the first term becomes 2x√x. Raising −2 by one gives −1, and −4 ÷ −1 = 4, so the second term becomes +4/x. So f(x) = 2x√x + 4/x + c.
Substitute (4, 10): 2 × 4 × 2 + 4/4 + c = 10, so 16 + 1 + c = 10 and c = −7. The answer is f(x) = 2x√x + 4/x − 7. Writing only c = −7 would lose the final mark: f(x) must be stated in full.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q1: integrating an expression with a negative power, as in Q1.
- January 2023, Q3: dividing a fraction into separate powers of x before integrating, as in Q2 and Q3.
- October 2024, Q3(a): a squared bracket over √x, expanded and divided before integrating, as in Q3.
- June 2024, Q10(b): f(x) from f′(x) and a point, with a minus sign in front of a fraction, as in Q4.
- October 2024, Q8: a normal fixes a constant, then integration and a point give f(x), as in Q7.
- January 2025, Q6: two conditions give simultaneous equations for a and b before integrating, as in Q8.
- January 2023, Q11: from f′′(x) with tangent information, integrating twice, as in Q9.
Where marks are lost
- Integrating before dividing. In January 2023, students who integrated each term of a fraction before dividing by the denominator scored no marks at all.
- The minus in front of a fraction. In June 2024 most students failed to carry the minus sign through when they simplified the fractional term.
- One constant for two integrations. In January 2023 it was very common to integrate f′′(x) twice and add a constant only at the end, and to use the point P where the gradient condition was needed.
- Normal or tangent gradient? In October 2024 the most common error was using the normal's gradient as if it were the tangent's.
- Stopping at c. A few students in October 2024 found the constant correctly but never wrote out f(x), and lost the final mark.
Common questions
Is integration of powers of x in the formula booklet?
No. Raising the power by one and dividing by the new power has to be learned.
Why can I not integrate the top and bottom of a fraction separately?
Integration does not work that way for a quotient. Divide every term by the denominator first, then integrate term by term.
Are definite integrals in Pure 1?
No. Definite integrals and areas under curves are in Pure 2; Pure 1 uses indefinite integrals with + c.
Where this chapter leads
- Pure 2 Chapter 8: Integration: definite integrals and areas under curves.
- Pure 2 Chapter 7: Differentiation: stationary points of curves like the ones built here.