The Split command splits the range of integration of v: ${\int }_a^{c}f\left(x\right)\ⅆx+{\int }_c^{b}f\left(x\right)\ⅆx={\int }_a^{b}f\left(x\right)\ⅆx$.

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.

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

$V≔\mathrm{Int}\left(\sin \left(x\right)\,x=1..2\mathrm{\pi }n-1\right)$

$V≔{\int }_1^{2\mathrm{\pi }n-1}\sin \left(x\right)\ⅆx$

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

${\int }_1^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{2\mathrm{\pi }}^{2\mathrm{\pi }n-1}\sin \left(x\right)\ⅆx$

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

${\int }_1^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{2\mathrm{\pi }}^{4\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{4\mathrm{\pi }}^{6\mathrm{\pi }}\sin \left(x\right)\ⅆx+{\int }_{6\mathrm{\pi }}^{2\mathrm{\pi }n-1}\sin \left(x\right)\ⅆx$

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

${\int }_1^{2\mathrm{\pi }}\sin \left(x\right)\ⅆx+\sum _{\mathrm{\_j}=1}^{n-2}\left({\int }_{2\mathrm{\pi }\mathrm{\_j}}^{2\mathrm{\pi }\left(\mathrm{\_j}+1\right)}\sin \left(x\right)\ⅆx\right)+{\int }_{2\mathrm{\pi }\left(n-1\right)}^{2\mathrm{\pi }n-1}\sin \left(x\right)\ⅆx$