About this chapter
Pure 3 integration extends the power rule of Pure 1 and Pure 2 to exponentials, reciprocals and trig functions. Almost every answer is exact, usually a single logarithm or a combination such as a + b ln 2, so the laws of logs from Chapter 5 matter as much as the integration.
This chapter covers the standard integrals, functions of (ax + b), the reverse chain rule for f′(x)/f(x), trig identities for sin²x, cos²x and tan²x, dividing improper fractions first, exact definite integrals, areas, unknown limits, and integrating a derivative found earlier. The later questions find the exact area of regions bounded by exponential and trig curves.
The ten questions
- Q1 (5 marks): integrate an exponential with a 4/x term, then 5/(2x − 3), then sec²4x.
- Q2 (5 marks): two exact definite integrals, one from 1 to e and one with an upper limit of ln 2.
- Q3 (5 marks): integrate (sin x + cos x)² and tan²x using identities.
- Q4 (5 marks): the reverse chain rule on 6x/(3x² + 1) and sin x/(2 + cos x), as a single log.
- Q5 (6 marks): divide an improper fraction first, then integrate it exactly as p + q ln r.
- Q6 (10 marks): rewrite a curve as x + 2 + 3/(x+2), the exact area under it, and its minimum point.
- Q7 (10 marks): expand (2 cos x − sin x)² with double angles, integrate it exactly, then a related integral by substitution.
- Q8 (10 marks): differentiate ln(sec x + tan x) to integrate sec x, show an integral of tan x is ln 2, and an area between curves.
- Q9 (10 marks): unknown limits: find k from an integral equal to ln 25, and p from a quadratic.
- Q10 (11 marks): a curve built from exponentials: where it crosses the axis at ln 2, its asymptote, an exact area, and a tangent.
Key skills tested
Standard integrals. e to the power x integrates to itself, 1/x to ln|x|, cos x to sin x, and sec²x to tan x, each plus a constant.
Functions of (ax + b). Integrate as usual, then divide by a. For example, 1/(2x − 3) integrates to ½ ln|2x − 3| + c.
The reverse chain rule. A fraction whose numerator is a multiple of the derivative of its denominator integrates to that multiple of ln|denominator|.
Trig identities. sin²x = ½(1 − cos 2x), cos²x = ½(1 + cos 2x), tan²x = sec²x − 1, and sin x cos x = ½ sin 2x.
Improper fractions. Divide first, then integrate each term. A term C/(x + a) gives C ln|x + a|.
Exact definite integrals. Substitute both limits, then use the laws of logarithms to write a single log. For example, 2 ln 5 − 2 ln 3 = 2 ln(5/3).
Areas. Integrate y between the limits. Below the x-axis the integral is negative, so take its size.
Unknown limits. Evaluate in terms of the unknown, solve, and reject values outside the domain.
Using "hence". A derivative found earlier gives an integral: if F′(x) = f(x), then f(x) integrates to F(x) + c.
Worked example
Question 1 from this chapter. Find (a) the integral of 3 e to the power 2x minus 4/x, (b) the integral of 5/(2x − 3), and (c) the integral of sec²(4x).
(a) e to the power 2x integrates to ½ e to the power 2x, so the first term gives 3/2 e to the power 2x. The 4/x term gives −4 ln|x|. The answer is 3/2 e to the power 2x − 4 ln|x| + c.
(b) 1/(2x − 3) integrates to ½ ln|2x − 3|, so 5/(2x − 3) gives 5/2 ln|2x − 3| + c.
(c) sec²(4x) integrates to ¼ tan 4x + c.
In every part, dividing by the coefficient of x is the step most often forgotten.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q2(b): an improper fraction, already divided, integrated to give a log term and an exact answer, as in Q5 and Q6(b).
- June 2024, Q6(b): an area made of the region under a curve plus a triangle, with the limits taken from a normal, like the areas in Q6 and Q10.
Where marks are lost
- The coefficient of the log term. In June 2024 coefficients were often confused, with answers such as (1/6) ln(x − 2) when the factor was wrong.
- Differentiating instead. A small minority differentiated the function, and could earn nothing in that part.
- The wrong limit. In June 2024 several used the wrong upper limit when the x coordinate of the point P, where the region changes, was needed.
- One integral for two regions. Trying to find a region made of a curve and a triangle with a single integral received no credit.
Common questions
When does an integral give ln?
When the integrand is 1/x, or a fraction whose numerator is a multiple of the derivative of its denominator.
How do I integrate sin²x?
Replace it with ½(1 − cos 2x) from the double-angle identity, then integrate term by term.
Should I use modulus signs in ln|x|?
Yes for indefinite integrals. With definite limits where the bracket is positive, brackets are enough, but be consistent.
Where this chapter leads
- Pure 4 Chapter 6: Integration: substitution, integration by parts, partial fractions and volumes of revolution.
- Pure 3 Chapter 8: Numerical Methods: roots and iteration for equations that integration and algebra cannot solve exactly.