Question

∫S 2^s ds integral por partes

Answer 1

$\int{s\,2^{s}ds}$


Integração por partes:

$\begin{array}{ll} u=s\;&\;du=ds\\ \\ dv=2^{s}\,ds\;&\;v=\dfrac{2^{s}}{\mathrm{\ell n\,}2} \end{array}\\ \\ \\ \int{u\,dv}=uv-\int{v\,du}\\ \\ \\ \int{s\,2^{s}ds}=\dfrac{s\,2^{s}}{\mathrm{\ell n\,}2}-\int{\dfrac{2^{s}}{\mathrm{\ell n\,}2}\,ds}\\ \\ \\ \int{s\,2^{s}ds}=\dfrac{s\,2^{s}}{\mathrm{\ell n\,}2}-\dfrac{1}{\mathrm{\ell n\,}2}\int{2^{s}\,ds}\\ \\ \\ \int{s\,2^{s}ds}=\dfrac{s\,2^{s}}{\mathrm{\ell n\,}2}-\dfrac{1}{\mathrm{\ell n\,}2}\cdot \left[\dfrac{2^{s}}{\mathrm{\ell n\,}2} \right ]+C\\ \\ \\ \int{s\,2^{s}ds}=\dfrac{s\,2^{s}}{\mathrm{\ell n\,}2}-\dfrac{2^{s}}{(\mathrm{\ell n\,}2)^{2}}+C$