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11.2PureCore

Integrating f(ax + b)

When the argument of a standard function is a linear expression ax + b, integrate as usual and divide by the coefficient a. This is the reverse of the chain rule for linear inner functions.

24 min Integration
Edexcel A level Maths: 11.2 Integrating f(ax + b)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Integrate functions of the form f(ax + b)
  • Divide by the coefficient of x
  • Apply to e, trig and power functions
  • Use in definite integrals
1

The rule

If ∫f(x) dx=F(x)+c\int f(x)\,dx = F(x) + c, then ∫f(ax+b) dx=1aF(ax+b)+c\int f(ax + b)\,dx = \dfrac{1}{a}F(ax + b) + c. You integrate the outer function and divide by aa.

∫f(ax+b) dx=1aF(ax+b)+c\int f(ax+b)\,dx = \frac{1}{a}F(ax+b) + c
Divide by the coefficient of x.
Example 1Find ∫e3x+1 dx\int e^{3x+1}\,dx.
1∫ex=ex\int e^x = e^x; here a=3a = 3.
213e3x+1+c\tfrac{1}{3}e^{3x+1} + c.
Answer
13e3x+1+c\tfrac13 e^{3x+1} + c

Tip — Only divide by a (the coefficient of x) — b does not affect the division.

Formula recap

∫eax+b dx=1aeax+b+c\int e^{ax+b}\,dx = \tfrac{1}{a}e^{ax+b} + c
Exponential.
∫cos⁡(ax+b) dx=1asin⁡(ax+b)+c\int \cos(ax+b)\,dx = \tfrac{1}{a}\sin(ax+b) + c
Cosine.

Common mistakes to avoid

Dividing by the constant b.
Divide by a, the coefficient of x.
Forgetting to divide by a entirely.
The 1/a factor is essential.

Key takeaways

  • ∫f(ax+b)dx = (1/a)F(ax+b) + c.
  • Integrate the outer function, divide by a.
  • b does not affect the division.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ∫e3x dx\int e^{3x}\,dx equals:

    • 13e3x+c\tfrac13 e^{3x} + c
    • 3e3x+c3e^{3x}+c
    • e3x+ce^{3x}+c
    • 13ex+c\tfrac13 e^{x}+c
    Show answer and explanation

    Answer: 13e3x+c\tfrac13 e^{3x} + c

    Divide by 3.

  2. ∫cos⁡(2x) dx\int \cos(2x)\,dx equals:

    • 12sin⁡2x+c\tfrac12\sin 2x + c
    • 2sin⁡2x+c2\sin 2x + c
    • sin⁡2x+c\sin 2x + c
    • −12sin⁡2x+c-\tfrac12\sin 2x + c
    Show answer and explanation

    Answer: 12sin⁡2x+c\tfrac12\sin 2x + c

    Divide by 2.

  3. In ∫f(ax+b)dx you divide by:

    • aa
    • bb
    • a+ba+b
    • abab
    Show answer and explanation

    Answer: aa

    Coefficient of x.

  4. ∫e3x+1 dx\int e^{3x+1}\,dx equals:

    • 13e3x+1+c\tfrac13 e^{3x+1} + c
    • e3x+1+ce^{3x+1}+c
    • 3e3x+1+c3e^{3x+1}+c
    • 14e3x+1+c\tfrac14 e^{3x+1}+c
    Show answer and explanation

    Answer: 13e3x+1+c\tfrac13 e^{3x+1} + c

    a = 3; b irrelevant to division.

  5. ∫(2x+1)3 dx\int (2x+1)^3\,dx equals:

    • 18(2x+1)4+c\tfrac18(2x+1)^4 + c
    • 14(2x+1)4+c\tfrac14(2x+1)^4 + c
    • (2x+1)4+c(2x+1)^4 + c
    • 12(2x+1)4+c\tfrac12(2x+1)^4+c
    Show answer and explanation

    Answer: 18(2x+1)4+c\tfrac18(2x+1)^4 + c

    Raise power /4, then /2.

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