Integral de $$$2^{5 x} 5^{- 2 x}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int 2^{5 x} 5^{- 2 x}\, dx$$$.
Solución
La entrada se reescribe: $$$\int{2^{5 x} 5^{- 2 x} d x}=\int{\left(\frac{32}{25}\right)^{x} d x}$$$.
Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=\frac{32}{25}$$$:
$${\color{red}{\int{\left(\frac{32}{25}\right)^{x} d x}}} = {\color{red}{\frac{\left(\frac{32}{25}\right)^{x}}{\ln{\left(\frac{32}{25} \right)}}}}$$
Por lo tanto,
$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{\ln{\left(\frac{32}{25} \right)}}$$
Simplificar:
$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln{\left(5 \right)} + 5 \ln{\left(2 \right)}}$$
Añade la constante de integración:
$$\int{\left(\frac{32}{25}\right)^{x} d x} = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln{\left(5 \right)} + 5 \ln{\left(2 \right)}}+C$$
Respuesta
$$$\int 2^{5 x} 5^{- 2 x}\, dx = \frac{\left(\frac{32}{25}\right)^{x}}{- 2 \ln\left(5\right) + 5 \ln\left(2\right)} + C$$$A