Split - Maple Help

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: ${{\int }}_{a}^{c}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x+{{\int }}_{c}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x={{\int }}_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}x$.
 • 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

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