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Integral de $$$2^{x} - 4^{x}$$$
Calculadora relacionada: Calculadora de integrales definidas e impropias
Tu entrada
Halla $$$\int \left(2^{x} - 4^{x}\right)\, dx$$$.
Solución
Integra término a término:
$${\color{red}{\int{\left(2^{x} - 4^{x}\right)d x}}} = {\color{red}{\left(\int{2^{x} d x} - \int{4^{x} d x}\right)}}$$
Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=2$$$:
$$- \int{4^{x} d x} + {\color{red}{\int{2^{x} d x}}} = - \int{4^{x} d x} + {\color{red}{\frac{2^{x}}{\ln{\left(2 \right)}}}}$$
Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=4$$$:
$$\frac{2^{x}}{\ln{\left(2 \right)}} - {\color{red}{\int{4^{x} d x}}} = \frac{2^{x}}{\ln{\left(2 \right)}} - {\color{red}{\frac{4^{x}}{\ln{\left(4 \right)}}}}$$
Por lo tanto,
$$\int{\left(2^{x} - 4^{x}\right)d x} = \frac{2^{x}}{\ln{\left(2 \right)}} - \frac{4^{x}}{\ln{\left(4 \right)}}$$
Simplificar:
$$\int{\left(2^{x} - 4^{x}\right)d x} = \frac{2 \cdot 2^{x} - 4^{x}}{2 \ln{\left(2 \right)}}$$
Añade la constante de integración:
$$\int{\left(2^{x} - 4^{x}\right)d x} = \frac{2 \cdot 2^{x} - 4^{x}}{2 \ln{\left(2 \right)}}+C$$
Respuesta
$$$\int \left(2^{x} - 4^{x}\right)\, dx = \frac{2 \cdot 2^{x} - 4^{x}}{2 \ln\left(2\right)} + C$$$A