<p><strong>Theorem</strong>\(\qquad\quad\) If&nbsp;\(u = g(x)\) is a differentiable function whose range is an interval \(I\) and \(f\) is continuous on \(I\), then</p>
<div class="content-box-placement content-box-placement--medium content-box-placement--align-center" style="display:flex;justify-content:center;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:60%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\int f(g(x))g'(x)\,\mathrm{d}x=\int f(u)\,\mathrm{d}u\)</p></div></div>
<p>This method of integration is helpful in reversing the chain rule.</p>
<p>Possible conditions seen when the method of substitution may be applicable:</p>
<ul class="bullet-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">A complicated expression with the "inside", \(g(x)\), and</div></li><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">the derivative of the complicated expression is seen somewhere in the integrand.</div></li></ul>

**Theorem**\(\qquad\quad\) If \(u = g(x)\) is a differentiable function whose range is an interval \(I\) and \(f\) is continuous on \(I\), then

\(\int f(g(x))g'(x)\,\mathrm{d}x=\int f(u)\,\mathrm{d}u\)

This method of integration is helpful in reversing the chain rule.

Possible conditions seen when the method of substitution may be applicable:

A complicated expression with the "inside", \(g(x)\), and

the derivative of the complicated expression is seen somewhere in the integrand.

Find \(\displaystyle\int x\left(x^2+1\right)^4\,\mathrm dx\). Let \(u=x^2+1\), so \(\mathrm du=2x\,\mathrm dx\) and \(x\,\mathrm dx=\dfrac12\mathrm du\).

\(\displaystyle\int x\left(x^2+1\right)^4\,\mathrm dx=\dfrac12\int u^4\,\mathrm du=\dfrac{u^5}{10}+C=\dfrac{\left(x^2+1\right)^5}{10}+C\)

**Exam check:** replace every \(x\), including \(\mathrm dx\), before integrating, and return to \(x\) at the end. Differentiate the answer: \(\dfrac{5\left(x^2+1\right)^4(2x)}{10}=x\left(x^2+1\right)^4\).

Integration by Substitution

Integration by Substitution