The IntegrationTools package is a set of programmer tools used for low level manipulation of definite and indefinite integrals.

Note: This package contains tools for manipulating the data structure only and do not ensure the validity of the operation being performed. For mathematical operations on integrals, use top-level commands such as combine, expand, etc., or the Student package.

At load time the IntegrationTools package defines three new types: Integral, DefiniteIntegral and IndefiniteIntegral, which can be used to access integrals involved in any given expression.

Each command in the IntegrationTools package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

The long form, IntegrationTools:-command is always available. The short form can be used after loading the package.

The following is a list of available commands.

To display the help page for a particular IntegrationTools command, see Getting Help with a Command in a Package.

$\mathrm{with}\left(\mathrm{IntegrationTools}\right)\:$

$v≔\mathrm{Int}\left(f\left(x\right)\,x=a..b\right)$

$v≔{\int }_a^{b}f\left(x\right)\ⅆx$

$\mathrm{type}\left(v\,\mathrm{Integral}\right)$

$\mathrm{true}$

$\mathrm{type}\left(v\,\mathrm{DefiniteIntegral}\right)$

$\mathrm{true}$

$\mathrm{type}\left(v\,\mathrm{IndefiniteIntegral}\right)$

$\mathrm{false}$

$\mathrm{GetIntegrand}\left(v\right)$

$f\left(x\right)$

$\mathrm{GetVariable}\left(v\right)$

$x$

$\mathrm{GetRange}\left(v\right)$

$a..b$

$v≔\mathrm{Int}\left(\sin \left(x\right)\,x=0..2\mathrm{\pi }n\right)$

$v≔{\int }_0^{2\mathrm{\pi }n}\sin \left(x\right)\ⅆx$

$\mathrm{Split}\left(v\,2\mathrm{\pi }\right)$

${\int }_0^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{2\mathrm{\pi }}^{2\mathrm{\pi }n}\sin \left(x\right)\ⅆx$

$\mathrm{Split}\left(v\,\left[2\mathrm{\pi }\,4\mathrm{\pi }\,6\mathrm{\pi }\right]\right)$

${\int }_0^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{2\mathrm{\pi }}^{4\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{4\mathrm{\pi }}^{6\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{6\mathrm{\pi }}^{2\mathrm{\pi }n}\sin \left(x\right)\ⅆx$

$\mathrm{Split}\left(v\,\left[2\mathrm{\pi }i\,i=1..n-1\right]\right)$

${\int }_0^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+\sum _{\mathrm{\_j}=1}^{n-2}\left({\int }_{2\mathrm{\pi }\mathrm{\_j}}^{2\mathrm{\pi }\left(\mathrm{\_j}+1\right)}\sin \left(x\right)\ⅆx\right)+{\int }_{2\mathrm{\pi }\left(n-1\right)}^{2\mathrm{\pi }n}\sin \left(x\right)\ⅆx$

$v≔\mathrm{Int}\left(\exp \left(x\right)\sin \left(x\right)\,x=a..b\right)$

$v≔{\int }_a^b{\ⅇ}^x\sin \left(x\right)\ⅆx$

$\mathrm{Parts}\left(v\,\sin \left(x\right)\right)$

${\ⅇ}^b\sin \left(b\right)-{\ⅇ}^a\sin \left(a\right)-\left({\int }_a^b{\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$\mathrm{Parts}\left(v\,\exp \left(x\right)\right)$

$-{\ⅇ}^b\cos \left(b\right)+{\ⅇ}^a\cos \left(a\right)-\left({\int }_a^b-{\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$v≔\mathrm{Int}\left(af\left(x\right)+bg\left(x\right)+ch\left(x\right)\,x=1..2\right)$

$v≔{\int }_1^2\left(af\left(x\right)+bg\left(x\right)+ch\left(x\right)\right)\ⅆx$

$w≔\mathrm{Expand}\left(v\right)$

$w≔a\left({\int }_1^{2}f\left(x\right)\ⅆx\right)+b\left({\int }_1^{2}g\left(x\right)\ⅆx\right)+c\left({\int }_1^{2}h\left(x\right)\ⅆx\right)$

$\mathrm{Combine}\left(w\right)$

${\int }_1^2\left(af\left(x\right)+bg\left(x\right)+ch\left(x\right)\right)\ⅆx$

$\mathrm{Combine}\left(\mathrm{Int}\left(f\left(x\right)\,x=a..b\right)+\mathrm{Int}\left(f\left(x\right)\,x=b..c\right)-\mathrm{Int}\left(f\left(x\right)\,x=a..d\right)\right)$

${\int }_d^{c}f\left(x\right)\ⅆx$