Integral+Rules

Integral Rules and Formulas  by: Veronica Manuel = = =Introduction= ​This page discusses various definite and indefinite integral properties as well as some other integral properties specific to certain types of functions (note that all of the rules that apply to indefinite integrals apply to definite integrals as well).  For the purpose of brevity, this page mentions but does not explain those integral properties that are deeply investigated by other people's wikipages in the integral section nor does it discuss other integral properties pertinent to the definitions of power series, velocity/displacement problems, and arc length and area applications. Doing so would extend beyond the grounds of integral "rules" and would go into all of the integral applications, definitions, and nuances. To learn all of that, an education in calculus is required. This page summarizes the most important integral rules and explanations are provided for the more complex integral rules. 

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**Known Derivative: for further explanation see Fundamental Theorem section below **  **or** **∴** where a and b are constants:
 * Rules for All Integrals **
 * Integral of a Constant Times a Function: ** If //f //is a function that can be integrated and a is a constant, then:
 * Integral of a Sum/Difference of Two Functions: ** If //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">f //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">and //g// are functions that can be integrated, then:

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">**Integration by Parts: If //u// and //v// are differentiable functions of //x,// then: <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; font-family: Times; font-size: medium; font-weight: normal;"> ** To evaluate this, these two criteria should be met.
 * Primary criterion:** //dv// should be something you can integrate.
 * Secondary criterion**: //u// should, if possible, be something that gets simpler (or at least not much more complicated) when it is differentiated.



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">Redirect to this website for a [|visual explanation] of this important integral property.
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Integral Mean Value Theorem: ** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">The integral mean value theorem (a corollary of the intermediate value theorem) states that a function continuous on an interval takes on its average value somewhere in the interval. More exactly, if [[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/1.gif align="absMiddle"]] is continuous on [[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/2.gif align="absMiddle"]], then there exists [[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/3.gif align="absMiddle"]] in [[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/4.gif align="absMiddle"]] such that [[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/5.gif align="absMiddle"]][[image:http://demonstrations.wolfram.com/IntegralMeanValueTheorem/HTMLImages/index.en/6.gif align="absMiddle"]].

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">**The Fundamental Theorem of Calculus:** Simply put: If //f// is an integrable function and, then:.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">This means, if, where //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">a // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;"> stands for a constant, and f is continuous in the neighborhood of //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">a // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">, then //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">g'(x) = f(x) // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">. <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">Proof: Let. Let h be and antiderivative of f. That is, let.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">
 * Rules for Definite Integrals **
 * Positive and Negative Integrands: The integral [[image:Screen_shot_2010-05-16_at_5.21.15_PM.png width="74" height="36"]] is positive if // f // ( // x // ) is positive for all values of // x // in [a,b] and negative if // f // ( // x // ) is negative for all values of // x // in [a,b], provided a < b.

Reversal of Limits of Integration:** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">**Integrals Between Symmetric Limits: If //f// is an odd function, then: If //f// is an even function, then: ** <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">**Upper Bounds for Integrals**: Suppose that the graph of one function is always below the graph of another. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">If //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">f // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">( //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">u // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">≤ //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">g // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">( //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">u // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">) for all //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">a // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">≤ //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">u // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">≤ //<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">b // <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">, then <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">**Absolute Value of an Integral:** If f is an integrable function and a < b, then: <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Sum of Integrals with the Same Integrand: **




 * Rules for Integrals of Specific Functions **

**Integral of the Natural Exponential Function:** If u// is a differentiable function, then: To differentiate, multiply by ln <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: sans-serif,Helvetica,sans-serif; font-size: 13px; line-height: 19px;">//<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">b. <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-style: normal;">To integrate, divide by ln <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">b // <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;"> and add <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: sans-serif,Helvetica,sans-serif; font-size: 13px; line-height: 19px;">//<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">C // <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">.
 * Integral of the Power function:** For any constant n <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: sans-serif; font-size: 13px; line-height: 19px;">≠ -1 and any differentiable function //u,
 * Derivative and Integral of Base-b Exponential Functions:** Given [[image:Screen_shot_2010-05-16_at_6.22.05_PM.png width="48" height="24"]]where //b// stands for a positive constant, //b// <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: sans-serif,Helvetica,sans-serif; font-size: 13px; line-height: 19px;">≠ 1, <span style="-webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; border-collapse: collapse; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 13px; line-height: 19px;">

where //x// <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">is a positive number.
 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Definition of the Natural Logarithm Function: **


 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">Integral of the Reciprocal Function: **

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif;">**Basic Integrals**



<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: small; line-height: normal;">[|An interactive look at some basic integral rules.]
 * [[image:AznBike.gif width="278" height="202"]] [[image:Screen_shot_2010-05-16_at_10.50.04_PM.png width="308" height="195" align="right"]]

__Works Cited__** http://www.calculusapplets.com/inttheorems.html http://www.mathwords.com/i/integral_rules.htm http://demonstrations.wolfram.com/IntegralMeanValueTheorem/ Calculus Concepts and Applications by Paul A. Foerster

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