applytoall

If there is more than one integral in the input, the applytoall option will perform integration by parts on each.

The Parts command performs integration by parts in an integral: $\int u\left(x\right)\mathrm{D}\left(v\right)\left(x\right)\ⅆx=u\left(v\right)v\left(x\right)-\left(\int v\left(x\right)\mathrm{D}\left(u\right)\left(x\right)\ⅆx\right)$. A similar transformation can be applied to definite integrals as well. By default the Parts command will apply the transformation to t only if it contains a single integral. In case of multiple integrals an error will be thrown. The Parts command can be forced to apply the same transformation to all integrals in t by setting the applytoall option to true.

The first parameter t is the integral.

The second parameter u is the u-term.

The third (optional) parameter v is the v-term. If this term is not specified it will be calculated from the first two parameters.

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

$V≔\mathrm{Int}\left(\exp \left(x\right)\sin \left(x\right)\,x\right)$

$V≔\int {\ⅇ}^x\sin \left(x\right)\ⅆx$

$\mathrm{Parts}\left(V\,\sin \left(x\right)\right)$

${\ⅇ}^x\sin \left(x\right)-\left(\int {\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$\mathrm{Parts}\left(V\,\exp \left(x\right)\right)$

$-{\ⅇ}^x\cos \left(x\right)-\left(\int -{\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$V≔\mathrm{Int}\left(\exp \left(x\right)\sin \left(x\right)\,x=a..b\right)$

$V≔{\int }_a^b{\ⅇ}^x\sin \left(x\right)\ⅆx$

$\mathrm{Parts}\left(V\,\sin \left(x\right)\right)$

${\ⅇ}^b\sin \left(b\right)-{\ⅇ}^a\sin \left(a\right)-\left({\int }_a^b{\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$\mathrm{Parts}\left(V\,\exp \left(x\right)\right)$

$-{\ⅇ}^b\cos \left(b\right)+{\ⅇ}^a\cos \left(a\right)-\left({\int }_a^b-{\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$V≔\mathrm{Int}\left(f\left(x\right)g\left(x\right)\,x=a..b\right)$

$V≔{\int }_a^{b}f\left(x\right)g\left(x\right)\ⅆx$

$\mathrm{Parts}\left(V\,f\left(x\right)\right)$

$\left(\int g\left(b\right)\ⅆb\right)f\left(b\right)-\left(\int g\left(a\right)\ⅆa\right)f\left(a\right)-\left({\int }_a^b\left(\int g\left(x\right)\ⅆx\right)\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)\ⅆx\right)$

$\mathrm{Parts}\left(V\,f\left(x\right)\,G\left(x\right)\right)$

$G\left(b\right)f\left(b\right)-G\left(a\right)f\left(a\right)-\left({\int }_a^{b}G\left(x\right)\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)\ⅆx\right)$

$U≔\mathrm{Int}\left(\exp \left(x\right)\sin \left(x\right)\,x\right)$

$U≔\int {\ⅇ}^x\sin \left(x\right)\ⅆx$

$V≔\mathrm{Int}\left(x^2\sin \left(x\right)\,x\right)$

$V≔\int x^2\sin \left(x\right)\ⅆx$

$W≔\mathrm{value}\left(V\right)$

$W≔-x^2\cos \left(x\right)+2\cos \left(x\right)+2x\sin \left(x\right)$

$\mathrm{Parts}\left(U\,\sin \left(x\right)\right)$

${\ⅇ}^x\sin \left(x\right)-\left(\int {\ⅇ}^x\cos \left(x\right)\ⅆx\right)$

$\mathrm{Parts}\left(U=W\,\sin \left(x\right)\right)$

${\ⅇ}^x\sin \left(x\right)-\left(\int {\ⅇ}^x\cos \left(x\right)\ⅆx\right)=-x^2\cos \left(x\right)+2\cos \left(x\right)+2x\sin \left(x\right)$

$\mathrm{Parts}\left(U+V\,\sin \left(x\right)\right)$

Error, (in IntegrationTools:-Parts) multiple integrals detected

$\mathrm{Parts}\left(U+V\,\sin \left(x\right)\,\mathrm{applytoall}=\mathrm{true}\right)$

${\ⅇ}^x\sin \left(x\right)-\left(\int {\ⅇ}^x\cos \left(x\right)\ⅆx\right)+\frac{x^3\sin \left(x\right)}{3}-\left(\int \frac{x^3\cos \left(x\right)}{3}\ⅆx\right)$