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IntegrationTools

 Change
 perform a change of variables in an integral

 Calling Sequence Change(v, c) Change(v, c, y)

Parameters

 v - expression involving definite or indefinite integrals c - a transformation or a list or set of transformations y - a list of the new variables

Description

 • The Change command performs a change of variables in the integrals involved in v.
 • The second parameter c is the transformation (or list of transformations or set of transformations) in the form x=f(u) (assuming x is the original variable and u is the new variable). Implicit transformations (g(x)=h(u)) can be given as long as it is possible to unambiguously rewrite it as an explicit transformation.
 • The third parameter y is required only if the number of new symbols in the transformation is not equal to the number of old variables in v.

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $V≔{{∫}}_{a}^{b}f\left({x}^{2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{{∫}}_{{a}}^{{b}}{f}{}\left({{x}}^{{2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (1)
 > $\mathrm{Change}\left(V,x={u}^{2}\right)$
 ${{∫}}_{\sqrt{{a}}}^{\sqrt{{b}}}{2}{}{f}{}\left({{u}}^{{4}}\right){}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}$ (2)

There are two new symbols in the transformation so the third argument is required.

 > $V≔{{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{{∫}}_{{a}}^{{b}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (3)
 > $\mathrm{Change}\left(V,x=u-c,u\right)$
 ${{∫}}_{{c}{+}{a}}^{{c}{+}{b}}{f}{}\left({u}{-}{c}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}$ (4)

Multivariate integral.

 > $V≔{{∫}}_{c}^{d}{{∫}}_{a}^{b}f\left({x}^{2}\right)g\left(y\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}y$
 ${V}{:=}{{∫}}_{{c}}^{{d}}{{∫}}_{{a}}^{{b}}{f}{}\left({{x}}^{{2}}\right){}{g}{}\left({y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}$ (5)
 > $\mathrm{Change}\left(V,\left\{x={u}^{2},y={v}^{2}\right\}\right)$
 ${{∫}}_{\sqrt{{c}}}^{\sqrt{{d}}}{2}{}\left({{∫}}_{\sqrt{{a}}}^{\sqrt{{b}}}{2}{}{f}{}\left({{u}}^{{4}}\right){}{g}{}\left({{v}}^{{2}}\right){}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right){}{v}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}$ (6)