Category:Integral Substitutions
This category contains examples of use of Integration by Substitution.
Let $\phi$ be a real function which has a derivative on the closed interval $\closedint a b$.
Let $I$ be an open interval which contains the image of $\closedint a b$ under $\phi$.
Let $f$ be a real function which is continuous on $I$.
Primitive
The primitive of $f$ can be evaluated by:
- $\ds \int \map f x \rd x = \int \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$
where $x = \map \phi u$.
Definite Integral
If $\map \phi a \le \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:
- $\ds \int_{\map \phi a}^{\map \phi b} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$
where $t = \map \phi u$.
If $\map \phi a > \map \phi b$, then the definite integral of $f$ from $a$ to $b$ can be evaluated by:
- $\ds - \int_{\map \phi b}^{\map \phi a} \map f t \rd t = \int_a^b \map f {\map \phi u} \dfrac \d {\d u} \map \phi u \rd u$
The technique of solving an integral in this manner is called integration by substitution.
Pages in category "Integral Substitutions"
The following 20 pages are in this category, out of 20 total.
I
P
- Primitive of Function of a x + b
- Primitive of Function of Arccosecant
- Primitive of Function of Arccosine
- Primitive of Function of Arccotangent
- Primitive of Function of Arcsecant
- Primitive of Function of Arcsine
- Primitive of Function of Arctangent
- Primitive of Function of Exponential Function
- Primitive of Function of Natural Logarithm
- Primitive of Function of Nth Root of a x + b
- Primitive of Function of Root of a squared minus x squared
- Primitive of Function of Root of a squared plus x squared
- Primitive of Function of Root of a x + b
- Primitive of Function of Root of x squared minus a squared