Integral de $$$e^{4 x} \sin{\left(5 x \right)}$$$

Halla $$$\int e^{4 x} \sin{\left(5 x \right)}\, dx$$$.

Para la integral $$$\int{e^{4 x} \sin{\left(5 x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\sin{\left(5 x \right)}$$$ y $$$\operatorname{dv}=e^{4 x} dx$$$.

Entonces $$$\operatorname{du}=\left(\sin{\left(5 x \right)}\right)^{\prime }dx=5 \cos{\left(5 x \right)} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{e^{4 x} d x}=\frac{e^{4 x}}{4}$$$ (los pasos pueden verse »).

Entonces,

$${\color{red}{\int{e^{4 x} \sin{\left(5 x \right)} d x}}}={\color{red}{\left(\sin{\left(5 x \right)} \cdot \frac{e^{4 x}}{4}-\int{\frac{e^{4 x}}{4} \cdot 5 \cos{\left(5 x \right)} d x}\right)}}={\color{red}{\left(\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \int{\frac{5 e^{4 x} \cos{\left(5 x \right)}}{4} d x}\right)}}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{5}{4}$$$ y $$$f{\left(x \right)} = e^{4 x} \cos{\left(5 x \right)}$$$:

$$\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - {\color{red}{\int{\frac{5 e^{4 x} \cos{\left(5 x \right)}}{4} d x}}} = \frac{e^{4 x} \sin{\left(5 x \right)}}{4} - {\color{red}{\left(\frac{5 \int{e^{4 x} \cos{\left(5 x \right)} d x}}{4}\right)}}$$

Para la integral $$$\int{e^{4 x} \cos{\left(5 x \right)} d x}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\cos{\left(5 x \right)}$$$ y $$$\operatorname{dv}=e^{4 x} dx$$$.

Entonces $$$\operatorname{du}=\left(\cos{\left(5 x \right)}\right)^{\prime }dx=- 5 \sin{\left(5 x \right)} dx$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{e^{4 x} d x}=\frac{e^{4 x}}{4}$$$ (los pasos pueden verse »).

La integral se convierte en

$$\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 {\color{red}{\int{e^{4 x} \cos{\left(5 x \right)} d x}}}}{4}=\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 {\color{red}{\left(\cos{\left(5 x \right)} \cdot \frac{e^{4 x}}{4}-\int{\frac{e^{4 x}}{4} \cdot \left(- 5 \sin{\left(5 x \right)}\right) d x}\right)}}}{4}=\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 {\color{red}{\left(\frac{e^{4 x} \cos{\left(5 x \right)}}{4} - \int{\left(- \frac{5 e^{4 x} \sin{\left(5 x \right)}}{4}\right)d x}\right)}}}{4}$$

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=- \frac{5}{4}$$$ y $$$f{\left(x \right)} = e^{4 x} \sin{\left(5 x \right)}$$$:

$$\frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 e^{4 x} \cos{\left(5 x \right)}}{16} + \frac{5 {\color{red}{\int{\left(- \frac{5 e^{4 x} \sin{\left(5 x \right)}}{4}\right)d x}}}}{4} = \frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 e^{4 x} \cos{\left(5 x \right)}}{16} + \frac{5 {\color{red}{\left(- \frac{5 \int{e^{4 x} \sin{\left(5 x \right)} d x}}{4}\right)}}}{4}$$

Hemos llegado a una integral que ya hemos visto.

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

$$\int{e^{4 x} \sin{\left(5 x \right)} d x} = \frac{e^{4 x} \sin{\left(5 x \right)}}{4} - \frac{5 e^{4 x} \cos{\left(5 x \right)}}{16} - \frac{25 \int{e^{4 x} \sin{\left(5 x \right)} d x}}{16}$$

Al resolverlo, obtenemos que

$$\int{e^{4 x} \sin{\left(5 x \right)} d x} = \frac{\left(4 \sin{\left(5 x \right)} - 5 \cos{\left(5 x \right)}\right) e^{4 x}}{41}$$

Por lo tanto,

$$\int{e^{4 x} \sin{\left(5 x \right)} d x} = \frac{\left(4 \sin{\left(5 x \right)} - 5 \cos{\left(5 x \right)}\right) e^{4 x}}{41}$$

Añade la constante de integración:

$$\int{e^{4 x} \sin{\left(5 x \right)} d x} = \frac{\left(4 \sin{\left(5 x \right)} - 5 \cos{\left(5 x \right)}\right) e^{4 x}}{41}+C$$

$$$\int e^{4 x} \sin{\left(5 x \right)}\, dx = \frac{\left(4 \sin{\left(5 x \right)} - 5 \cos{\left(5 x \right)}\right) e^{4 x}}{41} + C$$$A