Pearson Edexcel · A-Level · Mathematics

    Course version: 9MA0 · Version from Mar 2026

    Integrate xⁿ (excluding n = −1) and related sums, differences and constant…

    Full specification wording

    Integration: 8.2 Integrate xⁿ (excluding n = −1) and related sums, differences and constant multiples. Integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples.

    Standard integration reverses standard differentiation.

    Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .

    • Point 2 of 8
    • 3 key terms

    Study notes

    From the checked teaching notes for this specification point.

    The idea

    Standard integration reverses standard differentiation. For powers, ∫ xⁿ dx = x^(n+1)/(n+1) + c, valid for all n except n = −1; for example ∫ x³ dx = x⁴/4 + c. The exceptional case is ∫ 1/x dx = ln|x| + c, where the modulus keeps the logarithm defined for negative x. Exponential functions integrate to themselves with a scale factor: ∫ e^(kx) dx = (1/k)e^(kx) + c. Trigonometric functions give ∫ sin kx dx = −(1/k)cos kx + c and ∫ cos kx dx = (1/k)sin kx + c. Sums, differences and constant multiples are integrated term by term, so ∫ (3x² − 2 sin 2x + 5/x) dx = x³ + cos 2x + 5 ln|x| + c.

    What earns the marks

    • Applies the power rule ∫ xⁿ dx = x^(n+1)/(n+1) + c correctly for positive, negative and fractional n except n = −1.
    • Recognises the exceptional case and integrates 1/x as ln|x| + c rather than using the power rule.
    • Integrates e^(kx) as (1/k)e^(kx) + c, including the reciprocal scale factor.
    • Integrates sin kx as −(1/k)cos kx + c and cos kx as (1/k)sin kx + c with correct signs.
    • Integrates sums, differences and constant multiples term by term, carrying the constant of integration c.
    • Simplifies coefficients and exponents correctly, for example writing x^(5/2)/(5/2) as (2/5)x^(5/2).

    Common mistakes and how to put them right

    • Applying the power rule to 1/x and obtaining x⁰/0.

      Put it right by using ln|x| + c instead.

    • Forgetting the reciprocal factor when integrating e^(kx) or trigonometric functions.

      Put it right by dividing by k.

    • Omitting the constant of integration in an indefinite integral.

      Put it right by adding c to every indefinite result.

    Exam tips

    • Write each term on its own line before combining, so signs and coefficients are easy to check.
    • Differentiate your answer mentally to confirm it returns the original integrand.
    • Remember the modulus in ln|x| and the constant c in every indefinite integral.

    Try the lesson

    The start of the Integration: 8.2 Integrate xⁿ (excluding n = −1) and related sums, differences and constant multiples. Integrate e^(kx), 1/x, sin kx, cos kx and related sums, differences and constant multiples. lesson, made from checked teaching notes for this specification point.

    Have a go first

    Find the indefinite integral of x⁵ with respect to x.

    Listen

    Standard integration reverses standard differentiation. For powers, ∫ xⁿ dx = x^(n+1)/(n+1) + c, valid for all n except n = −1; for example ∫ x³ dx = x⁴/4 + c. The exceptional case is ∫ 1/x dx = ln|x| + c, where the modulus keeps the logarithm defined for negative x. Exponential functions integrate to themselves with a scale factor: ∫ e^(kx) dx = (1/k)e^(kx) + c. Trigonometric functions give ∫ sin kx dx = −(1/k)cos kx + c and ∫ cos kx dx = (1/k)sin kx + c. Sums, differences and constant multiples are integrated term by term, so ∫ (3x² − 2 sin 2x + 5/x) dx = x³ + cos 2x + 5 ln|x| + c. A strong answer earns marks for these points. Applies the power rule ∫ xⁿ dx = x^(n+1)/(n+1) + c correctly for positive, negative and fractional n except n = −1. Recognises the exceptional case and integrates 1/x as ln|x| + c rather than using the power rule. Integrates e^(kx) as (1/k)e^(kx) + c, including the reciprocal scale factor. Integrates sin kx as −(1/k)cos kx + c and cos kx as (1/k)sin kx + c with correct signs. A common slip: Applying the power rule to 1/x and obtaining x⁰/0. Instead, using ln|x| + c instead. Exam tip: Write each term on its own line before combining, so signs and coefficients are easy to check.

    See it

    A top-down flowchart. The top node reads 'Integrand: sum, difference or constant multiple' and points to a node reading 'Integrate term by term'. From that node, five branches lead to the term types: 'Power of x, n ≠ −1', '1/x', 'e^(kx)', 'sin kx' and 'cos kx'. Each term type then leads to its standard integral: the power of x leads to 'x^(n+1)/(n+1) + c'; 1/x leads to 'ln|x| + c'; e^(kx) leads to '(1/k)e^(kx) + c'; sin kx leads to '−(1/k)cos kx + c'; and cos kx leads to '(1/k)sin kx + c'.
    It shows how a sum, difference or constant multiple is integrated term by term, with the standard result for each type of term.

    Key words

    Say what each word means before you turn the card.

    Standard integrals as inverses of derivatives

    Try answering aloud or in your head before turning the card.

    The exceptional case of 1/x

    Try answering aloud or in your head before turning the card.

    Term-by-term integration of sums and differences

    Try answering aloud or in your head before turning the card.

    See every definition
    Standard integrals as inverses of derivatives
    Standard integration reverses standard differentiation, so powers, exponentials and trigonometric functions integrate back to related standard forms.
    The exceptional case of 1/x
    The power rule is valid for all n except n = −1, where 1/x integrates as ln|x| + c instead.
    Term-by-term integration of sums and differences
    Sums, differences and constant multiples are integrated term by term, carrying the constant of integration c.

    Label the diagram

    Some boxes have lost their names. Pick the right name for each number.

    The diagram again, with 3 boxes numbered instead of named. A top-down flowchart. The top node reads 'Integrand: sum, difference or constant multiple' and points to a node reading 'Integrate term by term'. From that node, five branches lead to the term types: 'Power of x, n ≠ −1', '1/x', 'e^(kx)', 'sin kx' and 'cos kx'. Each term type then leads to its standard integral: the power of x leads to 'x^(n+1)/(n+1) + c'; 1/x leads to 'ln|x| + c'; e^(kx) leads to '(1/k)e^(kx) + c'; sin kx leads to '−(1/k)cos kx + c'; and cos kx leads to '(1/k)sin kx + c'.

    • 1Box 1

    • 2Box 2

    • 3Box 3

    0 of 3 labelled

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