Integration by substitution: obey the given u

9 minA Level3.5.6

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can use a given substitution to turn a hard integral into an easy one, change the limits of a definite integral, and switch back to $x$ for an indefinite one.

The idea

Some integrals look scary in $x$ but become simple if you rename part of them as a new letter, $u$ .

For example, in $\displaystyle\int x\sqrt{1 + x^2}\,dx$ , call $1 + x^2$ by the name $u$ . The integral turns into $\tfrac12\displaystyle\int\sqrt u\,du$ , which you can do with the power rule.

In P3 the question tells you which substitution to use. Your job is to carry it out correctly.

Key to the symbols

Symbol	Meaning
$u$	the new variable, defined by the substitution, e.g. $u = 1 + x^2$
$\dfrac{du}{dx}$	the derivative of $u$ with respect to $x$
$du$ , $dx$	tiny changes in $u$ and $x$ ; $du = \dfrac{du}{dx}\,dx$
$u'(x)$ , $g'(x)$	other ways to write a derivative, e.g. $u'(x) = \dfrac{du}{dx}$

The method

Differentiate the substitution. For $u = 1 + x^2$ : $\;\dfrac{du}{dx} = 2x$ .
Rearrange to get $dx$ (or a piece like $x\,dx$ ) in terms of $du$ : $\;du = 2x\,dx$ , so $x\,dx = \tfrac12\,du$ .
Replace everything. Every $x$ and the $dx$ must go. Only $u$ and $du$ are left.
Integrate in $u$ .
Finish:
- For a definite integral, change the limits to $u$ -values (before integrating) and use those.
- For an indefinite integral, swap $u$ back to $x$ at the end and add $+\,C$ .

The step people forget: step 2. You can't just change $dx$ to $du$ . They differ by the factor $\dfrac{du}{dx}$ .

Worked example: Example 1: indefinite, switching back to x

Use the substitution $u = 1 + x^2$ to find $\displaystyle\int x\sqrt{1 + x^2}\,dx$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 3 questions above to continue.