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Integrating standard functions

AS Maths (Edexcel) · Edexcel 8MA0 · P8.2 · Pure mathematics › Integration

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Revision notes

You need to integrate powers of \(x\), including negative and fractional powers other than \(x^{-1}\), with sums, differences and constant multiples. These results appear in almost every integration question on Paper 1.

Key facts

  • \(\int x^n\,\mathrm{d}x = \dfrac{x^{n+1}}{n+1} + c\), \(n \neq -1\)

Notes

Powers of x

  • \(\int x^n\,\mathrm{d}x = \dfrac{x^{n+1}}{n+1} + c\) works for every \(n\) except \(-1\), including fractional and negative powers.
  • Rewrite first: \(\sqrt{x} = x^{\frac{1}{2}}\), \(\dfrac{3}{x^2} = 3x^{-2}\), \(\dfrac{1}{2\sqrt{x}} = \dfrac{1}{2}x^{-\frac{1}{2}}\). Expand brackets and split fractions that have a single term underneath.

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Exam technique

  • An integral with no limits needs \(+\,c\) in the final answer.
  • In a 'show that', write the square-bracket line and the substituted values: the final line alone earns nothing.
  • Know the basic results by heart, and check before the exam which extra results the formula booklet gives you.

What each grade needs

What you need to be able to do, from the first marks up to the top grade.

  1. Grade E
    Integrate powers of x, including fractional powersRewrite roots and fractions as powers first, e.g. \(\dfrac{2}{\sqrt{x}} = 2x^{-\frac{1}{2}}\), which integrates to \(4x^{\frac{1}{2}} + c\)

Sample questions

Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.

Question 1Easy4 marks
In this question \(x \gt 0\).
Find \(\int \left(-4x^{3} + 3x^{2} - 8x + 6\sqrt{x}\right) \,\mathrm{d}x\), giving each term in its simplest form.[4]
Show the answer and mark scheme
Answer: \(-x^{4} + x^{3} - 4x^{2} + 4x^{\frac{3}{2}} + c\)
  • M1 for \(x^{n} \to x^{n + 1}\) for at least one term
  • A1 for two terms correct (may be unsimplified)
  • A1 for three terms correct
  • A1 for a fully correct, simplified answer including \(+ c\)

Worked solution: Writing each term as a power of \(x\): \(-4x^{3} + 3x^{2} - 8x + 6x^{\frac{1}{2}}\).
\(\int \left(-4x^{3} + 3x^{2} - 8x + 6x^{\frac{1}{2}}\right) \,\mathrm{d}x = -x^{4} + x^{3} - 4x^{2} + 4x^{\frac{3}{2}} + c\)

Question 2Medium7 marks
\(f(x) = \frac{(3x + 2)(x - 4)}{\sqrt{x}}, \quad x \gt 0\)
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
(a) Write \(f(x)\) in the form \(Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}\), where \(A\), \(B\) and \(C\) are constants to be found.[2]
(b) Hence find \(\int f(x) \,\mathrm{d}x\).[3]
(c) Hence find the exact value of \(\int_{1}^{4} f(x) \,\mathrm{d}x\).[2]
Show the answer and mark scheme
(a) Answer: \(A = 3\), \(B = -10\), \(C = -8\)
  • M1 for expanding the numerator and dividing each term by the denominator
  • A1 for \(A = 3\), \(B = -10\), \(C = -8\)

Worked solution: Expanding and dividing term by term: \(f(x) = 3x^{\frac{3}{2}} - 10x^{\frac{1}{2}} - 8x^{-\frac{1}{2}}\), so \(A = 3\), \(B = -10\) and \(C = -8\).

(b) Answer: \(\frac{6}{5}x^{\frac{5}{2}} - \frac{20}{3}x^{\frac{3}{2}} - 16x^{\frac{1}{2}} + c\)
  • M1 for \(x^{n} \to x^{n + 1}\) for at least one term
  • A1 for two terms correct
  • A1 for \(\frac{6}{5}x^{\frac{5}{2}} - \frac{20}{3}x^{\frac{3}{2}} - 16x^{\frac{1}{2}} + c\)

Worked solution: \(\int f(x) \,\mathrm{d}x = \frac{6}{5}x^{\frac{5}{2}} - \frac{20}{3}x^{\frac{3}{2}} - 16x^{\frac{1}{2}} + c\)

(c) Answer: \(-\frac{382}{15}\)
  • M1 for substituting the limits 4 and 1 into their answer to part (b) and subtracting
  • A1 for \(-\frac{382}{15}\)

Worked solution: \(\left[\frac{6}{5}x^{\frac{5}{2}} - \frac{20}{3}x^{\frac{3}{2}} - 16x^{\frac{1}{2}}\right]_{1}^{4} = \left(-\frac{704}{15}\right) - \left(-\frac{322}{15}\right) = -\frac{382}{15}\).

Question 3Hard6 marks
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
(a) Find \(\frac{\mathrm{d}}{\mathrm{d}x}\left((x^{2} + 9)^{\frac{3}{2}}\right)\).[2]
(b) Hence find \(\int 2x\sqrt{x^{2} + 9} \,\mathrm{d}x\).[2]
(c) Hence find the exact value of \(\int_{0}^{4} 2x\sqrt{x^{2} + 9} \,\mathrm{d}x\).[2]
Show the answer and mark scheme
(a) Answer: \(3x(x^{2} + 9)^{\frac{1}{2}}\)
  • M1 for \(\lambda x(x^{2} + 9)^{\frac{1}{2}}\)
  • A1 for \(3x(x^{2} + 9)^{\frac{1}{2}}\)

Worked solution: By the chain rule: \(\frac{3}{2}(x^{2} + 9)^{\frac{1}{2}} \times 2x = 3x(x^{2} + 9)^{\frac{1}{2}}\).

(b) Answer: \(\frac{2}{3}(x^{2} + 9)^{\frac{3}{2}} + c\)
  • M1 for using part (a) to obtain \(\lambda(x^{2} + 9)^{\frac{3}{2}}\)
  • A1 for \(\frac{2}{3}(x^{2} + 9)^{\frac{3}{2}} + c\)

Worked solution: From part (a), \(\int 3x\sqrt{x^{2} + 9} \,\mathrm{d}x = (x^{2} + 9)^{\frac{3}{2}} + c\), so \(\int 2x\sqrt{x^{2} + 9} \,\mathrm{d}x = \frac{2}{3}(x^{2} + 9)^{\frac{3}{2}} + c\).

(c) Answer: \(\frac{196}{3}\)
  • M1 for substituting the limits into their answer to part (b)
  • A1 for \(\frac{196}{3}\)

Worked solution: \(\left[\frac{2}{3}(x^{2} + 9)^{\frac{3}{2}}\right]_{0}^{4} = \frac{2}{3}\left(25^{\frac{3}{2}} - 9^{\frac{3}{2}}\right) = \frac{2}{3}(125 - 27) = \frac{196}{3}\).

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