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IntegrationTools

 Parts
 perform integration by parts

 Calling Sequence Parts(t, u) Parts(t, u, v) Parts(t, u, applytoall) Parts(t, u, v, applytoall)

Parameters

 t - expression containing definite or indefinite integrals u - u-term v - v-term

Options

 • applytoall
 If there is more than one integral in the input, the applytoall option will perform integration by parts on each.

Description

 • The Parts command performs integration by parts in an integral: ${∫}u\left(x\right)\mathrm{D}\left(v\right)\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x=u\left(v\right)v\left(x\right)-\left({∫}v\left(x\right)\mathrm{D}\left(u\right)\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x\right)$. A similar transformation can be applied to definite integrals as well. By default the Parts command will apply the transformation to t only if it contains a single integral. In case of multiple integrals an error will be thrown. The Parts command can be forced to apply the same transformation to all integrals in t by setting the applytoall option to true.
 • The first parameter t is the integral.
 • The second parameter u is the u-term.
 • The third (optional) parameter v is the v-term. If this term is not specified it will be calculated from the first two parameters.

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $V≔{∫}{ⅇ}^{x}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{∫}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (1)
 > $\mathrm{Parts}\left(V,\mathrm{sin}\left(x\right)\right)$
 ${{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){-}\left({∫}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (2)
 > $\mathrm{Parts}\left(V,{ⅇ}^{x}\right)$
 ${-}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right){-}\left({∫}\left({-}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (3)

Definite integral.

 > $V≔{{∫}}_{a}^{b}{ⅇ}^{x}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{{∫}}_{{a}}^{{b}}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (4)
 > $\mathrm{Parts}\left(V,\mathrm{sin}\left(x\right)\right)$
 ${{ⅇ}}^{{b}}{}{\mathrm{sin}}{}\left({b}\right){-}{{ⅇ}}^{{a}}{}{\mathrm{sin}}{}\left({a}\right){-}\left({{∫}}_{{a}}^{{b}}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (5)
 > $\mathrm{Parts}\left(V,{ⅇ}^{x}\right)$
 ${-}{{ⅇ}}^{{b}}{}{\mathrm{cos}}{}\left({b}\right){+}{{ⅇ}}^{{a}}{}{\mathrm{cos}}{}\left({a}\right){-}\left({{∫}}_{{a}}^{{b}}\left({-}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (6)

Specifying both u and v.

 > $V≔{{∫}}_{a}^{b}f\left(x\right)g\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{{∫}}_{{a}}^{{b}}{f}{}\left({x}\right){}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (7)
 > $\mathrm{Parts}\left(V,f\left(x\right)\right)$
 $\underset{{x}{→}{{b}}^{{-}}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({∫}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{f}{}\left({x}\right){-}\left(\underset{{x}{→}{{a}}^{{+}}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({∫}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}{f}{}\left({x}\right)\right){-}\left({{∫}}_{{a}}^{{b}}\left({∫}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (8)
 > $\mathrm{Parts}\left(V,f\left(x\right),G\left(x\right)\right)$
 ${G}{}\left({b}\right){}{f}{}\left({b}\right){-}{G}{}\left({a}\right){}{f}{}\left({a}\right){-}\left({{∫}}_{{a}}^{{b}}{G}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (9)

Dealing with multiple integrals

 > $U≔{∫}{ⅇ}^{x}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${U}{:=}{∫}{{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (10)
 > $V≔{∫}{x}^{2}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{∫}{{x}}^{{2}}{}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (11)
 > $W≔\mathrm{value}\left(V\right)$
 ${W}{:=}{-}{{x}}^{{2}}{}{\mathrm{cos}}{}\left({x}\right){+}{2}{}{\mathrm{cos}}{}\left({x}\right){+}{2}{}{x}{}{\mathrm{sin}}{}\left({x}\right)$ (12)
 > $\mathrm{Parts}\left(U,\mathrm{sin}\left(x\right)\right)$
 ${{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){-}\left({∫}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (13)
 > $\mathrm{Parts}\left(U=W,\mathrm{sin}\left(x\right)\right)$
 ${{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){-}\left({∫}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){=}{-}{{x}}^{{2}}{}{\mathrm{cos}}{}\left({x}\right){+}{2}{}{\mathrm{cos}}{}\left({x}\right){+}{2}{}{x}{}{\mathrm{sin}}{}\left({x}\right)$ (14)
 > $\mathrm{Parts}\left(U+V,\mathrm{sin}\left(x\right)\right)$
 > $\mathrm{Parts}\left(U+V,\mathrm{sin}\left(x\right),\mathrm{applytoall}=\mathrm{true}\right)$
 ${{ⅇ}}^{{x}}{}{\mathrm{sin}}{}\left({x}\right){-}\left({∫}{{ⅇ}}^{{x}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}\frac{{1}}{{3}}{}{{x}}^{{3}}{}{\mathrm{sin}}{}\left({x}\right){-}\left({∫}\frac{{1}}{{3}}{}{{x}}^{{3}}{}{\mathrm{cos}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (15)