Integral de $$$e^{5 x} \sin{\left(3 x \right)}$$$

Halla $$$\int e^{5 x} \sin{\left(3 x \right)}\, dx$$$.

Para la integral $$$\int{e^{5 x} \sin{\left(3 x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\sin{\left(3 x \right)}$$$ y $$$\operatorname{dv}=e^{5 x} dx$$$.

Entonces $$$\operatorname{du}=\left(\sin{\left(3 x \right)}\right)^{\prime }dx=3 \cos{\left(3 x \right)} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{e^{5 x} d x}=\frac{e^{5 x}}{5}$$$ (los pasos pueden verse »).

La integral se convierte en

$${\color{red}{\int{e^{5 x} \sin{\left(3 x \right)} d x}}}={\color{red}{\left(\sin{\left(3 x \right)} \cdot \frac{e^{5 x}}{5}-\int{\frac{e^{5 x}}{5} \cdot 3 \cos{\left(3 x \right)} d x}\right)}}={\color{red}{\left(\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \int{\frac{3 e^{5 x} \cos{\left(3 x \right)}}{5} d x}\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{3}{5}$$$ y $$$f{\left(x \right)} = e^{5 x} \cos{\left(3 x \right)}$$$:

$$\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - {\color{red}{\int{\frac{3 e^{5 x} \cos{\left(3 x \right)}}{5} d x}}} = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - {\color{red}{\left(\frac{3 \int{e^{5 x} \cos{\left(3 x \right)} d x}}{5}\right)}}$$

Para la integral $$$\int{e^{5 x} \cos{\left(3 x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\cos{\left(3 x \right)}$$$ y $$$\operatorname{dv}=e^{5 x} dx$$$.

Entonces $$$\operatorname{du}=\left(\cos{\left(3 x \right)}\right)^{\prime }dx=- 3 \sin{\left(3 x \right)} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{e^{5 x} d x}=\frac{e^{5 x}}{5}$$$ (los pasos pueden verse »).

Por lo tanto,

$$\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 {\color{red}{\int{e^{5 x} \cos{\left(3 x \right)} d x}}}}{5}=\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 {\color{red}{\left(\cos{\left(3 x \right)} \cdot \frac{e^{5 x}}{5}-\int{\frac{e^{5 x}}{5} \cdot \left(- 3 \sin{\left(3 x \right)}\right) d x}\right)}}}{5}=\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 {\color{red}{\left(\frac{e^{5 x} \cos{\left(3 x \right)}}{5} - \int{\left(- \frac{3 e^{5 x} \sin{\left(3 x \right)}}{5}\right)d x}\right)}}}{5}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=- \frac{3}{5}$$$ y $$$f{\left(x \right)} = e^{5 x} \sin{\left(3 x \right)}$$$:

$$\frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25} + \frac{3 {\color{red}{\int{\left(- \frac{3 e^{5 x} \sin{\left(3 x \right)}}{5}\right)d x}}}}{5} = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25} + \frac{3 {\color{red}{\left(- \frac{3 \int{e^{5 x} \sin{\left(3 x \right)} d x}}{5}\right)}}}{5}$$

Hemos llegado a una integral que ya hemos visto.

Así, hemos obtenido la siguiente ecuación simple con respecto a la integral:

$$\int{e^{5 x} \sin{\left(3 x \right)} d x} = \frac{e^{5 x} \sin{\left(3 x \right)}}{5} - \frac{3 e^{5 x} \cos{\left(3 x \right)}}{25} - \frac{9 \int{e^{5 x} \sin{\left(3 x \right)} d x}}{25}$$

Al resolverlo, obtenemos que

$$\int{e^{5 x} \sin{\left(3 x \right)} d x} = \frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}$$

Por lo tanto,

$$\int{e^{5 x} \sin{\left(3 x \right)} d x} = \frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}$$

Añade la constante de integración:

$$\int{e^{5 x} \sin{\left(3 x \right)} d x} = \frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34}+C$$

$$$\int e^{5 x} \sin{\left(3 x \right)}\, dx = \frac{\left(5 \sin{\left(3 x \right)} - 3 \cos{\left(3 x \right)}\right) e^{5 x}}{34} + C$$$A