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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

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

Definite integrals will be given a common variable of integration

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