Changes

Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>

Intergation has the following properties<ref>[http://www.sosmath.com/calculus/integ/integ02/integ02.html Properties of Intergrals]</ref>

<h3>1. Additive Distribution</h3>

<h3>Additive Distribution</h3>

If an integral contains two functions added together, it may be rewritten as two separate integrals, each containing one function.

If an integral contains two functions added together, it may be rewritten as two separate integrals, each containing one function.

*<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>

*<math>\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx</math>

<h3>2. Constant Multiplicative Distribution</h3>

<h3>Constant Multiplicative Distribution</h3>

If an integral contains a function multiplied by a constant, it may be rewritten as the constant times the integral of the function.

If an integral contains a function multiplied by a constant, it may be rewritten as the constant times the integral of the function.

*<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>

*<math>\int_{a}^{b}cf(x)dx=c\int_{a}^{b}f(x)dx</math>

<h3>3. Area</h3>

<h3>Area</h3>

Any integral that contains the same upper and lower bounds is equal to zero.

Any integral that contains the same upper and lower bounds is equal to zero.

*<math>\int_{a}^{a}f(x)dx=0</math>  

*<math>\int_{a}^{a}f(x)dx=0</math>  

<h3>4. Separation</h3>

<h3>Separation</h3>

An integral may be rewritten as the sum of two integrals with adjacent bounds.

An integral may be rewritten as the sum of two integrals with adjacent bounds.

*<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>

*<math>\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx</math>, <math>c\in(a,b)</math>

<h3>5. Additive Inversion</h3>

<h3>Additive Inversion</h3>

Any integral is equal to the additive inverse of the same integral with reversed upper and lower bounds.

Any integral is equal to the additive inverse of the same integral with reversed upper and lower bounds.

*<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>

*<math>\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx</math>