257
edits
Changes
Jump to navigation
Jump to search
Line 17:
Line 17: + + + + + + + + + +
→Properties of integrals
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>
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>
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>
Any integral that contains the same upper and lower limit 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>
An integral may be rewritten as the sum of two integrals with adjacent limits.
*<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>
Any integral is equal to the additive inverse of the same integral with reversed upper and lower limits.
*<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>