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Solo haré la que aparentemente parece difícil
![\displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\int\limits_{0}^{1}\int\limits_{1}^{2}xe^x\cdot \left[\dfrac{1}{y}\;dy\right]\, dx\\\\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\int\limits_{0}^{1}xe^x\left[\int\limits_{1}^{2} \dfrac{1}{y}\;dy\right]\, dx
\displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\int\limits_{0}^{1}\int\limits_{1}^{2}xe^x\cdot \left[\dfrac{1}{y}\;dy\right]\, dx\\\\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\int\limits_{0}^{1}xe^x\left[\int\limits_{1}^{2} \dfrac{1}{y}\;dy\right]\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7Dxe%5Ex%5Ccdot+%5Cleft%5B%5Cdfrac%7B1%7D%7By%7D%5C%3Bdy%5Cright%5D%5C%2C+dx%5C%5C%5C%5C+%5C%5C%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cint%5Climits_%7B0%7D%5E%7B1%7Dxe%5Ex%5Cleft%5B%5Cint%5Climits_%7B1%7D%5E%7B2%7D+%5Cdfrac%7B1%7D%7By%7D%5C%3Bdy%5Cright%5D%5C%2C+dx%0A)
![\displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left(\int\limits_{0}^{1}xe^x\, dx\right) \cdot \left(\int\limits_{1}^{2} \dfrac{1}{y}\;dy\right)\\ \\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left(\int\limits_{0}^{1}x\, d(e^x)\right) \cdot \left.\left(\ln y\right)\right|_{y=1}^{y=2} \displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left(\int\limits_{0}^{1}xe^x\, dx\right) \cdot \left(\int\limits_{1}^{2} \dfrac{1}{y}\;dy\right)\\ \\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left(\int\limits_{0}^{1}x\, d(e^x)\right) \cdot \left.\left(\ln y\right)\right|_{y=1}^{y=2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cleft%28%5Cint%5Climits_%7B0%7D%5E%7B1%7Dxe%5Ex%5C%2C+dx%5Cright%29+%5Ccdot+%5Cleft%28%5Cint%5Climits_%7B1%7D%5E%7B2%7D+%5Cdfrac%7B1%7D%7By%7D%5C%3Bdy%5Cright%29%5C%5C+%5C%5C+%5C%5C%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cleft%28%5Cint%5Climits_%7B0%7D%5E%7B1%7Dx%5C%2C+d%28e%5Ex%29%5Cright%29+%5Ccdot+%5Cleft.%5Cleft%28%5Cln+y%5Cright%29%5Cright%7C_%7By%3D1%7D%5E%7By%3D2%7D)
![\displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left.\left(xe^x- \int e^x\, dx\right)\right|_{0}^{1} \cdot \ln 2\\ \\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left.\left(xe^x- e^x\right)\right|_{0}^{1} \cdot \ln 2\\ \\ \\
\boxed{\boxed{\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\ln 2}} \displaystyle
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left.\left(xe^x- \int e^x\, dx\right)\right|_{0}^{1} \cdot \ln 2\\ \\ \\
\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\left.\left(xe^x- e^x\right)\right|_{0}^{1} \cdot \ln 2\\ \\ \\
\boxed{\boxed{\int\limits_{0}^{1}\int\limits_{1}^{2}\dfrac{xe^x}{y}\;dy\, dx=\ln 2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cleft.%5Cleft%28xe%5Ex-+%5Cint+e%5Ex%5C%2C+dx%5Cright%29%5Cright%7C_%7B0%7D%5E%7B1%7D+%5Ccdot+%5Cln+2%5C%5C+%5C%5C+%5C%5C%0A%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cleft.%5Cleft%28xe%5Ex-+e%5Ex%5Cright%29%5Cright%7C_%7B0%7D%5E%7B1%7D+%5Ccdot+%5Cln+2%5C%5C+%5C%5C+%5C%5C%0A%5Cboxed%7B%5Cboxed%7B%5Cint%5Climits_%7B0%7D%5E%7B1%7D%5Cint%5Climits_%7B1%7D%5E%7B2%7D%5Cdfrac%7Bxe%5Ex%7D%7By%7D%5C%3Bdy%5C%2C+dx%3D%5Cln+2%7D%7D)
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