Integrating composite functions

Covers AS91579

Reversing the chain rule

When the "inside" of a function is linear (of the form $ax + b$ ), you can integrate by reversing the chain rule.
Integrate as if the bracket were a single $x$ , then divide by the derivative of the inside (the coefficient $a$ ):

$\int (ax + b)^n\,dx = \frac{(ax + b)^{\,n+1}}{a(n + 1)} + c \qquad (n \neq -1)$

The exponential and trig versions

The same "divide by $a$ " idea applies to exponentials and trig functions with a linear inside:

$\int e^{ax + b}\,dx = \frac{1}{a}e^{ax + b} + c$

$\int \sin(ax + b)\,dx = -\frac{1}{a}\cos(ax + b) + c \qquad \int \cos(ax + b)\,dx = \frac{1}{a}\sin(ax + b) + c$

Worked ExampleReverse chain rule

Find $\displaystyle\int (2x + 1)^5\,dx$ .

Step 1 — Integrate as if the bracket were a single variable

Add 1 to the power and divide by the new power (6):

$\frac{(2x + 1)^6}{6}$

Step 2 — Divide by the derivative of the inside

The inside $2x + 1$ has derivative 2, so divide by 2 as well:

$\int (2x + 1)^5\,dx = \frac{(2x + 1)^6}{6 \times 2} + c = \frac{(2x + 1)^6}{12} + c$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find $\displaystyle\int (3x + 2)^4\,dx$ .

Merit

Find $\displaystyle\int e^{4x}\,dx$ and $\displaystyle\int \cos(3x)\,dx$ .

Excellence

Find $\displaystyle\int \left(2e^{-x} + \sin(2x)\right)\,dx$ .