<p><strong>Algebraic functions</strong></p>
<p>In the table below, \(A, B\) and \(n\) are constants, and \(\mathrm{f}\) is a function; \(C\) is an arbitrary constant.</p>
<div class="content-block content-block--table"><div class="table-container"><table><caption>Algebraic Functions: power of variable, and of a function</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">\[\frac{\mathrm{d}}{\mathrm{d}x}(x^{n+1})=(n+1)x^n\]</td><td style="text-align: left; --cell-justify: flex-start">\[\displaystyle\int x^n \, \mathrm{d}x = \frac{x^{n+1}}{n+1} + C, \text{ where } n \neq -1\]</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\[\frac{\mathrm{d}}{\mathrm{d}x}(Ax+B)^{n+1}=A(n+1)(Ax+B)^n\]</td><td style="text-align: left; --cell-justify: flex-start">\[\displaystyle\int (Ax+B)^n \, \mathrm{d}x = \frac{(Ax+B)^{n+1}}{A(n+1)} + C, \text{ where } n \neq -1\]</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\[\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{f}(x)]^{n+1}=(n+1)[\mathrm{f}(x)]^n[\mathrm{f}'(x)]\]</td><td style="text-align: left; --cell-justify: flex-start">\[\displaystyle\int [\mathrm{f}(x)]^n[\mathrm{f}'(x)] \, \mathrm{d}x = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + C, \text{ where } n \neq -1\]</td></tr></tbody></table></div></div>

In the table below, \(A, B\) and \(n\) are constants, and \(\mathrm{f}\) is a function; \(C\) 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
\[\frac{\mathrm{d}}{\mathrm{d}x}(x^{n+1})=(n+1)x^n\]	\[\displaystyle \displaystyle\int x^n \, \mathrm{d}x = \frac{x^{n+1}}{n+1} + C, \text{ where}\hspace{0.5em} n \neq -1\]
\[\frac{\mathrm{d}}{\mathrm{d}x}(Ax+B)^{n+1}=A(n+1)(Ax+B)^n\]	\[\displaystyle \displaystyle\int (Ax+B)^n \, \mathrm{d}x = \frac{(Ax+B)^{n+1}}{A(n+1)} + C, \text{ where}\hspace{0.5em} n \neq -1\]
\[\frac{\mathrm{d}}{\mathrm{d}x}[\mathrm{f}(x)]^{n+1}=(n+1)[\mathrm{f}(x)]^n[\mathrm{f}'(x)]\]	\[\displaystyle \displaystyle\int [\mathrm{f}(x)]^n[\mathrm{f}'(x)] \, \mathrm{d}x = \frac{[\mathrm{f}(x)]^{n+1}}{n+1} + C, \text{ where}\hspace{0.5em} n \neq -1\]