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 Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.2 Integrating f(ax + b)Watch the full walkthrough 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

The rule

If $\int f (x) d x = F (x) + c$ , then $\int f (a x + b) d x = \frac{1}{a} F (a x + b) + c$ . You integrate the outer function and divide by $a$ .

\int f (a x + b) d x = \frac{1}{a} F (a x + b) + c

Divide by the coefficient of x.

\int e^{x} = e^{x}

; here

a = 3

\frac{1}{3} e^{3 x + 1} + c

Answer

\frac{1}{3} e^{3 x + 1} + c

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

Formula recap

\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c

Exponential.

\int cos (a x + b) d x = \frac{1}{a} sin (a x + 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.

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