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IntegrationTools

 Split
 split the range of integration

 Calling Sequence Split(v, c)

Parameters

 v - definite or indefinite integral c - splitting point(s)

Description

 • The Split command splits the range of integration of v: ${{∫}}_{a}^{c}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x+{{∫}}_{c}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={{∫}}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$.
 • The second parameter c is a splitting point or a list of splitting points. Alternatively, a list of the form [f(i), i = m..n] can be specified. In this case [seq(f(i), i=m..n)] will be used as the splitting points if m and n are constants.  If at least one of m or n is symbolic, the points f(m), f(m+1), ... , f(n-1), f(n) will be used and the result is correct as long as (m-n) is an integer.

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $V≔{{∫}}_{1}^{2\mathrm{π}n-1}\mathrm{sin}\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$
 ${V}{:=}{{∫}}_{{1}}^{{2}{}{\mathrm{π}}{}{n}{-}{1}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (1)
 > $\mathrm{Split}\left(V,2\mathrm{π}\right)$
 ${{∫}}_{{1}}^{{2}{}{\mathrm{π}}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{∫}}_{{2}{}{\mathrm{π}}}^{{2}{}{\mathrm{π}}{}{n}{-}{1}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (2)
 > $\mathrm{Split}\left(V,\left[2\mathrm{π},4\mathrm{π},6\mathrm{π}\right]\right)$
 ${{∫}}_{{1}}^{{2}{}{\mathrm{π}}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{∫}}_{{2}{}{\mathrm{π}}}^{{4}{}{\mathrm{π}}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{∫}}_{{4}{}{\mathrm{π}}}^{{6}{}{\mathrm{π}}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{{∫}}_{{6}{}{\mathrm{π}}}^{{2}{}{\mathrm{π}}{}{n}{-}{1}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (3)
 > $\mathrm{Split}\left(V,\left[2\mathrm{π}i,i=1..n-1\right]\right)$
 ${{∫}}_{{1}}^{{2}{}{\mathrm{π}}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\sum }_{{\mathrm{_j}}{=}{1}}^{{n}{-}{2}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({{∫}}_{{2}{}{\mathrm{π}}{}{\mathrm{_j}}}^{{2}{}{\mathrm{π}}{}\left({\mathrm{_j}}{+}{1}\right)}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{{∫}}_{{2}{}{\mathrm{π}}{}\left({n}{-}{1}\right)}^{{2}{}{\mathrm{π}}{}{n}{-}{1}}{\mathrm{sin}}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (4)