Combine - Maple Help
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IntegrationTools

 Combine
 combine integrals using linearity

 Calling Sequence Combine(v)

Parameters

 v - expression

Description

 • The Combine command combines integrals using linearity.
 • The parameter v is any expression involving definite or indefinite integrals. Definite integrals will have their variable of integration renamed to facilitate combination.
 • For indefinite integrals, this command is equivalent to calling combine(v, int).

Examples

 > $\mathrm{with}\left(\mathrm{IntegrationTools}\right):$
 > $v≔\mathrm{Int}\left(af\left(x\right)+bg\left(x\right)+ch\left(x\right),x=1..2\right)$
 ${v}{≔}{{\int }}_{{1}}^{{2}}\left({a}{}{f}{}\left({x}\right){+}{b}{}{g}{}\left({x}\right){+}{c}{}{h}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (1)
 > $w≔\mathrm{Expand}\left(v\right)$
 ${w}{≔}{a}{}\left({{\int }}_{{1}}^{{2}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{b}{}\left({{\int }}_{{1}}^{{2}}{g}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){+}{c}{}\left({{\int }}_{{1}}^{{2}}{h}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)$ (2)
 > $\mathrm{Combine}\left(w\right)$
 ${{\int }}_{{1}}^{{2}}\left({a}{}{f}{}\left({x}\right){+}{b}{}{g}{}\left({x}\right){+}{c}{}{h}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (3)

Definite integrals will be given a common variable of integration

 > $\mathrm{Combine}\left(\mathrm{Int}\left(f\left(x\right),x=a..b\right)+\mathrm{Int}\left(f\left(y\right),y=b..c\right)-\mathrm{Int}\left(f\left(z\right),z=a..d\right)\right)$
 ${{\int }}_{{d}}^{{c}}{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (4)
 > $\mathrm{Combine}\left(\mathrm{Int}\left(f\left(x\right),x\right)+\mathrm{Int}\left(g\left(x\right),x\right)\right)$
 ${\int }\left({f}{}\left({x}\right){+}{g}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (5)