<p><strong>Generalised powers</strong></p>
<p>In the table below, \(\mathrm{f}\) is a function, and \(n\) is a constant; and \(C\) is an arbitrary constant.</p>
<div class="content-block content-block--table"><div class="table-container"><table><caption>Generalised powers</caption><thead><tr><th style="text-align: left; --cell-justify: flex-start">Differentiation</th><th style="text-align: left; --cell-justify: flex-start">Integration</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\[\frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{[\mathrm{f}(x)]^{n+1}}{n+1} \right) = [\mathrm{f}(x)]^n \mathrm{f}'(x)\]</td><td style="text-align: left; --cell-justify: flex-start">\[\displaystyle\int [\mathrm{f}(x)]^n \mathrm{f}'(x) \, \mathrm{d}x = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + C \quad (n \neq -1)\]</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\[\frac{\mathrm{d}}{\mathrm{d}x} (\ln \mathrm{f}(x)) = \frac{\mathrm{f}'(x)}{\mathrm{f}(x)}\]</td><td style="text-align: left; --cell-justify: flex-start">\[\displaystyle\int \frac{\mathrm{f}'(x)}{\mathrm{f}(x)} \, \mathrm{d}x = \ln |\mathrm{f}(x)| + C\]</td></tr></tbody></table></div></div>

In the table below, \(\mathrm{f}\) is a function, and \(n\) is a constant; and \(C\) is an arbitrary constant.

Integration by the Reverse Chain Rule

Techniques of Integration

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Techniques of Integration - Integration by the Reverse Chain Rule

Techniques of Integration - Integration by the Reverse Chain Rule

Differentiation	Integration
\[\frac{\mathrm{d}}{\mathrm{d}x} \left( \frac{[\mathrm{f}(x)]^{n+1}}{n+1} \right) = [\mathrm{f}(x)]^n \mathrm{f}'(x)\]	\[\displaystyle \displaystyle\int [\mathrm{f}(x)]^n \mathrm{f}'(x) \, \mathrm{d}x = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + C \quad (n \neq -1)\]
\[\frac{\mathrm{d}}{\mathrm{d}x} (\ln \mathrm{f}(x)) = \frac{\mathrm{f}'(x)}{\mathrm{f}(x)}\]	\[\displaystyle \displaystyle\int \frac{\mathrm{f}'(x)}{\mathrm{f}(x)} \, \mathrm{d}x = \ln \|\mathrm{f}(x)\| + C\]