[tex]\boxed{\begin{matrix}\text{Integra\c c\~ao por partes} :\\\\ \displaystyle \sf \int u\ dv =u\cdot v -\int v\ du \end{matrix}}\ \\\\\\ \displaystyle \sf temos :\\\\ \int e^{x}sen(x)dx \\\\\\ \text{Fa\c camos } : \\\\ u = e^{x}\to du = e^{x}dx \\\\ dv = sen(x)dx \to v = -cos(x) \\\\ Da{\'i}}: \\\\ \int e^{x}sen(x)dx = \int u\ dv = u\cdot v -\int v\ du \\\\\\ u\cdot v -\int v\ du=-e^{x}cos(x)-\int -e^{x}cos(x)dx \\\\\\ \int e^{x}sen(x)dx = -e^{x}cos(x)+\int e^{x}cos(x)dx \\\\[/tex]

[tex]\displaystyle \sf \text{Fa\c camos} : \\\\ m = e^{x} \to dm=e^{x} \ \ ;\ \ dn = cos(x)dx \to n = sen(x) \\\\ Da{\'i}}: \\\\ \int e^{x}sen(x)dx = -e^{x}cos(x)+\int e^{x}cos(x)dx \\\\\\ \int e^{x}sen(x)dx = -e^{x}cos(x)+\left[\int m\ dn \right] \\\\\\ \int e^{x}sen(x)dx =-e^{x}cos(x)+\left[m\cdot n -\int n\ dm \right] \\\\\\ \int e^{x}sen(x)dx = -e^{x}cos(x)+e^{x}sen(x)-\int e^{x}sen(x)dx[/tex]

[tex]\displaystyle \sf \int e^{x}sen(x)dx +\int e^{x}sen(x)dx = -e^{x}cos(x)+e^{x}sen(x) \\\\\\ 2\int e^{x}sen(x)dx = -e^{x}cos(x)+e^{x}sen(x) +C\\\\\\ \Large\boxed{\sf \ \int e^{x}sen(x)dx = \frac{-e^{x}cos(x)}{2}+\frac{e^{x}sen(x)}{2}+C \ }\checkmark[/tex]

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