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<h1 class="center">Integration Rules</h1>
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<h2>Integration</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/integral-area.gif" alt="integral area" height="171" width="195"></p>
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<p><a href="integration-introduction.html">Integration</a> can be used to find areas, volumes, central points and many useful things. It is often used to find the <b>area underneath the graph of a function and the x-axis</b>.</p>
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<p>The first rule to know is that integrals and <a href="derivatives-introduction.html">derivatives</a> are opposites!</p>
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<p class="center"><img src="images/integral-vs-derivative.svg" alt="integral vs derivative" height="122" width="222"><br>
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Sometimes we can work out an integral,<br>
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because we know a matching derivative.</p>
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<h3>Integration Rules</h3>
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<p>Here are the most useful rules, with <a href="#examples">examples below</a>:</p>
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<div class="beach">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th>Common Functions</th>
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<th align="center" width="130">Function</th>
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<th align="center" width="130">Integral</th>
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</tr>
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<tr>
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<td>Constant</td>
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<td style="text-align:center;"><span class="int">∫</span>a dx</td>
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<td style="text-align:center;">ax + C</td>
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</tr>
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<tr>
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<td>Variable</td>
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<td style="text-align:center;"><span class="int">∫</span>x dx</td>
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<td style="text-align:center;">x<sup>2</sup>/2 + C</td>
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</tr>
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<tr>
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<td>Square</td>
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<td style="text-align:center;"><span class="int">∫</span>x<sup>2</sup> dx</td>
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<td style="text-align:center;">x<sup>3</sup>/3 + C</td>
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</tr>
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<tr>
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<td>Reciprocal</td>
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<td style="text-align:center;"><span class="int">∫</span>(1/x) dx</td>
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<td style="text-align:center;">ln|x| + C</td>
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</tr>
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<tr>
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<td>Exponential</td>
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<td style="text-align:center;"><span class="int">∫</span>e<sup>x</sup> dx</td>
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<td style="text-align:center;">e<sup>x</sup> + C</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"><span class="int">∫</span>a<sup>x</sup> dx</td>
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<td style="text-align:center;">a<sup>x</sup>/ln(a) + C</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"><span class="int">∫</span>ln(x) dx</td>
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<td style="text-align:center;">x ln(x) − x + C</td>
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</tr>
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<tr>
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<td>Trigonometry (x in <a href="../geometry/radians.html">radians</a>)</td>
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<td style="text-align:center;"><span class="int">∫</span>cos(x) dx</td>
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<td style="text-align:center;">sin(x) + C</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"><span class="int">∫</span>sin(x) dx</td>
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<td style="text-align:center;">-cos(x) + C</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"><span class="int">∫</span>sec<sup>2</sup>(x) dx</td>
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<td style="text-align:center;">tan(x) + C</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<th>Rules</th>
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<th align="center">Function<br>
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</th>
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<th align="center">Integral<br>
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</th>
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</tr>
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<tr>
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<td>Multiplication by constant</td>
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<td style="text-align:center;"><span class="int">∫</span>cf(x) dx</td>
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<td style="text-align:center;">c<span class="int">∫</span>f(x) dx</td>
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</tr>
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<tr>
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<td>Power Rule (n≠−1)</td>
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<td style="text-align:center;"><span class="int">∫</span>x<sup>n</sup> dx</td>
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<td style="text-align:center;"><span class="intbl"><em>x<sup>n+1</sup></em><strong>n+1</strong></span> + C</td>
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</tr>
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<tr>
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<td>Sum Rule</td>
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<td style="text-align:center;"><span class="int">∫</span>(f + g) dx</td>
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<td style="text-align:center;"><span class="int">∫</span>f dx + <span class="int">∫</span>g dx</td>
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</tr>
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<tr>
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<td>Difference Rule</td>
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<td style="text-align:center;"><span class="int">∫</span>(f - g) dx</td>
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<td style="text-align:center;"><span class="int">∫</span>f dx - <span class="int">∫</span>g dx</td>
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</tr>
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<tr>
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<td>Integration by Parts</td>
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<td colspan="2" align="center">See <a href="integration-by-parts.html">Integration by Parts</a></td>
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</tr>
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<tr>
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<td>Substitution Rule</td>
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<td colspan="2" align="center">See <span class="center"><a href="integration-by-substitution.html">Integration by Substitution</a></span></td>
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</tr>
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</tbody></table>
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</div>
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<h2><a name="examples"></a>Examples</h2>
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<div class="example">
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<h3>Example: what is the integral of sin(x) ?</h3>
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<p>From the table above it is listed as being <b>−cos(x) + C</b></p>
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<p>It is written as:</p>
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<p class="center large"><span class="int">∫</span>sin(x) dx = −cos(x) + C</p>
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</div>
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<div class="example">
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<h3>Example: what is the integral of 1/x ?</h3>
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<p>From the table above it is listed as being <b>ln|x| + C</b></p>
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<p>It is written as:</p>
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<p class="center large"><span class="int">∫</span>(1/x) dx = ln|x| + C</p>
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<p>The vertical bars <b>||</b> either side of <b>x</b> mean <a href="../numbers/absolute-value.html">absolute value</a>, because we don't want to give negative values to the <a href="../sets/function-logarithmic.html">natural logarithm</a> function <b>ln</b>.</p>
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</div>
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<h3>Power Rule</h3>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>x<sup>3</sup> dx ?</h3>
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<p>The question is asking "what is the integral of x<sup>3 </sup>?"</p>
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<p>We can use the Power Rule, where n=3:</p>
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<p class="center large"><span class="int">∫</span>x<sup>n</sup> dx = <span class="intbl"><em>x<sup>n+1</sup></em><strong>n+1</strong></span> + C</p>
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<p class="center large"><span class="int">∫</span>x<sup>3 </sup>dx = <span class="intbl"><em>x<sup>4</sup></em><strong>4</strong></span> + C</p>
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</div>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>√x dx ?</h3>
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<p>√x is also <b>x<sup>0.5</sup></b></p>
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<p>We can use the Power Rule, where n=0.5:</p>
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<p class="center large"><span class="int">∫</span>x<sup>n</sup> dx = <span class="intbl"><em>x<sup>n+1</sup></em><strong>n+1</strong></span> + C</p>
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<p class="center large"><span class="int">∫</span>x<sup>0.5</sup> dx = <span class="intbl"><em>x<sup>1.5</sup></em><strong>1.5</strong></span> + C</p>
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</div>
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<h3>Multiplication by constant</h3>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>6x<sup>2</sup> dx ?</h3>
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<p>We can move the 6 outside the integral:</p>
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<p class="center large"><span class="int">∫</span>6x<sup>2</sup> dx = 6<span class="int">∫</span>x<sup>2</sup> dx</p>
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<p>And now use the Power Rule on <span class="center large">x<sup>2</sup></span>:</p>
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<p class="center large">= 6 <span class="intbl"><em>x<sup>3</sup></em><strong>3</strong></span> + C</p>
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<p>Simplify:</p>
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<p class="center large">= 2x<sup>3</sup> + C</p>
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</div>
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<h3>Sum Rule</h3>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>(cos x + x) dx ?</h3>
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<p>Use the Sum Rule:</p>
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<p class="center large"><span class="int">∫</span>(cos x + x) dx = <span class="int">∫</span>cos x dx + <span class="int">∫</span>x dx</p>
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<p>Work out the integral of each (using table above):</p>
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<p class="center large">= sin x + x<sup>2</sup>/2 + C</p>
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</div>
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<h3>Difference Rule</h3>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>(e<sup>w</sup> − 3) dw ?</h3>
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<p>Use the Difference Rule:</p>
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<p class="center large"><span class="int">∫</span>(e<sup>w</sup> − 3) dw =<span class="int">∫</span>e<sup>w</sup> dw − <span class="int">∫</span>3 dw</p>
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<p>Then work out the integral of each (using table above):</p>
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<p class="center large">= e<sup>w</sup> − 3w + C</p>
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</div>
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<h3>Sum, Difference, Constant Multiplication And Power Rules</h3>
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<div class="example">
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<h3>Example: What is <span class="int">∫</span>(8z + 4z<sup>3</sup> − 6z<sup>2</sup>) dz ?</h3>
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<p>Use the Sum and Difference Rule:</p>
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<p class="center large"><span class="int">∫</span>(8z + 4z<sup>3</sup> − 6z<sup>2</sup>) dz =<span class="int">∫</span>8z dz + <span class="int">∫</span>4z<sup>3</sup> dz − <span class="int">∫</span>6z<sup>2</sup> dz</p>
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<p>Constant Multiplication:</p>
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<p class="center large">= 8<span class="int">∫</span>z dz + 4<span class="int">∫</span>z<sup>3</sup> dz − 6<span class="int">∫</span>z<sup>2</sup> dz</p>
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<p>Power Rule:</p>
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<p class="center large">= 8z<sup>2</sup>/2 + 4z<sup>4</sup>/4 − 6z<sup>3</sup>/3 + C</p>
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<p>Simplify:</p>
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<p class="center large">= 4z<sup>2</sup> + z<sup>4</sup> − 2z<sup>3</sup> + C</p>
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</div>
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<h3>Integration by Parts</h3>
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<p>See <a href="integration-by-parts.html">Integration by Parts</a></p>
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<h3>Substitution Rule</h3>
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<p>See <span class="center"><a href="integration-by-substitution.html">Integration by Substitution</a></span></p>
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<p> </p>
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<h2>Final Advice</h2>
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<ul>
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<li>Get plenty of practice</li>
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<li>Don't forget the <b>dx</b> (or dz, etc)</li>
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<li>Don't forget the <b>+ C</b></li>
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</ul>
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<p> </p>
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<div class="questions">6834, 6835, 6836, 6837, 6838, 6839, 6840, 6841, 6842, 6843</div>
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<div class="related">
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<a href="integration-introduction.html">Integration</a>
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<a href="index.html">Calculus Index</a>
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