Integral de ((3*2^(x)-2*3^(x))/5^x) dx
Solución
Solución detallada
Vuelva a escribir el integrando:
3 ⋅ 2 x − 2 ⋅ 3 x 5 x = 3 ⋅ 2 x 5 − x − 2 ⋅ 3 x 5 − x \frac{3 \cdot 2^{x} - 2 \cdot 3^{x}}{5^{x}} = 3 \cdot 2^{x} 5^{- x} - 2 \cdot 3^{x} 5^{- x} 5 x 3 ⋅ 2 x − 2 ⋅ 3 x = 3 ⋅ 2 x 5 − x − 2 ⋅ 3 x 5 − x
Integramos término a término:
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ 3 ⋅ 2 x 5 − x d x = 3 ∫ 2 x 5 − x d x \int 3 \cdot 2^{x} 5^{- x}\, dx = 3 \int 2^{x} 5^{- x}\, dx ∫ 3 ⋅ 2 x 5 − x d x = 3 ∫ 2 x 5 − x d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
2 x − 5 x log ( 5 ) + 5 x log ( 2 ) \frac{2^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(2 \right)}} − 5 x l o g ( 5 ) + 5 x l o g ( 2 ) 2 x
Por lo tanto, el resultado es: 3 ⋅ 2 x − 5 x log ( 5 ) + 5 x log ( 2 ) \frac{3 \cdot 2^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(2 \right)}} − 5 x l o g ( 5 ) + 5 x l o g ( 2 ) 3 ⋅ 2 x
La integral del producto de una función por una constante es la constante por la integral de esta función:
∫ ( − 2 ⋅ 3 x 5 − x ) d x = − 2 ∫ 3 x 5 − x d x \int \left(- 2 \cdot 3^{x} 5^{- x}\right)\, dx = - 2 \int 3^{x} 5^{- x}\, dx ∫ ( − 2 ⋅ 3 x 5 − x ) d x = − 2 ∫ 3 x 5 − x d x
No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
3 x − 5 x log ( 5 ) + 5 x log ( 3 ) \frac{3^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(3 \right)}} − 5 x l o g ( 5 ) + 5 x l o g ( 3 ) 3 x
Por lo tanto, el resultado es: − 2 ⋅ 3 x − 5 x log ( 5 ) + 5 x log ( 3 ) - \frac{2 \cdot 3^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(3 \right)}} − − 5 x l o g ( 5 ) + 5 x l o g ( 3 ) 2 ⋅ 3 x
El resultado es: 3 ⋅ 2 x − 5 x log ( 5 ) + 5 x log ( 2 ) − 2 ⋅ 3 x − 5 x log ( 5 ) + 5 x log ( 3 ) \frac{3 \cdot 2^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(2 \right)}} - \frac{2 \cdot 3^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(3 \right)}} − 5 x l o g ( 5 ) + 5 x l o g ( 2 ) 3 ⋅ 2 x − − 5 x l o g ( 5 ) + 5 x l o g ( 3 ) 2 ⋅ 3 x
Ahora simplificar:
2 5 − x ( 1 0 x log ( 27 125 ) + 1 5 x log ( 25 4 ) ) log ( 2 5 ) log ( 3 5 ) \frac{25^{- x} \left(10^{x} \log{\left(\frac{27}{125} \right)} + 15^{x} \log{\left(\frac{25}{4} \right)}\right)}{\log{\left(\frac{2}{5} \right)} \log{\left(\frac{3}{5} \right)}} l o g ( 5 2 ) l o g ( 5 3 ) 2 5 − x ( 1 0 x l o g ( 125 27 ) + 1 5 x l o g ( 4 25 ) )
Añadimos la constante de integración:
2 5 − x ( 1 0 x log ( 27 125 ) + 1 5 x log ( 25 4 ) ) log ( 2 5 ) log ( 3 5 ) + c o n s t a n t \frac{25^{- x} \left(10^{x} \log{\left(\frac{27}{125} \right)} + 15^{x} \log{\left(\frac{25}{4} \right)}\right)}{\log{\left(\frac{2}{5} \right)} \log{\left(\frac{3}{5} \right)}}+ \mathrm{constant} l o g ( 5 2 ) l o g ( 5 3 ) 2 5 − x ( 1 0 x l o g ( 125 27 ) + 1 5 x l o g ( 4 25 ) ) + constant
Respuesta:
2 5 − x ( 1 0 x log ( 27 125 ) + 1 5 x log ( 25 4 ) ) log ( 2 5 ) log ( 3 5 ) + c o n s t a n t \frac{25^{- x} \left(10^{x} \log{\left(\frac{27}{125} \right)} + 15^{x} \log{\left(\frac{25}{4} \right)}\right)}{\log{\left(\frac{2}{5} \right)} \log{\left(\frac{3}{5} \right)}}+ \mathrm{constant} l o g ( 5 2 ) l o g ( 5 3 ) 2 5 − x ( 1 0 x l o g ( 125 27 ) + 1 5 x l o g ( 4 25 ) ) + constant
Respuesta (Indefinida)
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| x x x x
| 3*2 - 2*3 2*3 3*2
| ----------- dx = C - --------------------- + ---------------------
| x x x x x
| 5 5 *log(3) - 5 *log(5) 5 *log(2) - 5 *log(5)
|
/
∫ 3 ⋅ 2 x − 2 ⋅ 3 x 5 x d x = 3 ⋅ 2 x − 5 x log ( 5 ) + 5 x log ( 2 ) − 2 ⋅ 3 x − 5 x log ( 5 ) + 5 x log ( 3 ) + C \int \frac{3 \cdot 2^{x} - 2 \cdot 3^{x}}{5^{x}}\, dx = \frac{3 \cdot 2^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(2 \right)}} - \frac{2 \cdot 3^{x}}{- 5^{x} \log{\left(5 \right)} + 5^{x} \log{\left(3 \right)}} + C ∫ 5 x 3 ⋅ 2 x − 2 ⋅ 3 x d x = − 5 x log ( 5 ) + 5 x log ( 2 ) 3 ⋅ 2 x − − 5 x log ( 5 ) + 5 x log ( 3 ) 2 ⋅ 3 x + C
Gráfica
0.00 1.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0 2
log(5) 6*log(2) 3*log(3) 2*log(2) 6*log(3)
------------------------------------------------------- - --------------------------------------------------------------- - ------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------------
2 2 2 2 2
log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) 5*log (5) - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3) log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) 5*log (5) - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3)
− 3 log ( 3 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 − 6 log ( 2 ) − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 + 6 log ( 3 ) − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 + 2 log ( 2 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 + log ( 5 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 - \frac{3 \log{\left(3 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} - \frac{6 \log{\left(2 \right)}}{- 5 \log{\left(3 \right)} \log{\left(5 \right)} - 5 \log{\left(2 \right)} \log{\left(5 \right)} + 5 \log{\left(2 \right)} \log{\left(3 \right)} + 5 \log{\left(5 \right)}^{2}} + \frac{6 \log{\left(3 \right)}}{- 5 \log{\left(3 \right)} \log{\left(5 \right)} - 5 \log{\left(2 \right)} \log{\left(5 \right)} + 5 \log{\left(2 \right)} \log{\left(3 \right)} + 5 \log{\left(5 \right)}^{2}} + \frac{2 \log{\left(2 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} + \frac{\log{\left(5 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} − − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 3 log ( 3 ) − − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 6 log ( 2 ) + − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 6 log ( 3 ) + − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 2 log ( 2 ) + − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 log ( 5 )
=
log(5) 6*log(2) 3*log(3) 2*log(2) 6*log(3)
------------------------------------------------------- - --------------------------------------------------------------- - ------------------------------------------------------- + ------------------------------------------------------- + ---------------------------------------------------------------
2 2 2 2 2
log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) 5*log (5) - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3) log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) log (5) + log(2)*log(3) - log(2)*log(5) - log(3)*log(5) 5*log (5) - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3)
− 3 log ( 3 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 − 6 log ( 2 ) − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 + 6 log ( 3 ) − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 + 2 log ( 2 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 + log ( 5 ) − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 - \frac{3 \log{\left(3 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} - \frac{6 \log{\left(2 \right)}}{- 5 \log{\left(3 \right)} \log{\left(5 \right)} - 5 \log{\left(2 \right)} \log{\left(5 \right)} + 5 \log{\left(2 \right)} \log{\left(3 \right)} + 5 \log{\left(5 \right)}^{2}} + \frac{6 \log{\left(3 \right)}}{- 5 \log{\left(3 \right)} \log{\left(5 \right)} - 5 \log{\left(2 \right)} \log{\left(5 \right)} + 5 \log{\left(2 \right)} \log{\left(3 \right)} + 5 \log{\left(5 \right)}^{2}} + \frac{2 \log{\left(2 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} + \frac{\log{\left(5 \right)}}{- \log{\left(3 \right)} \log{\left(5 \right)} - \log{\left(2 \right)} \log{\left(5 \right)} + \log{\left(2 \right)} \log{\left(3 \right)} + \log{\left(5 \right)}^{2}} − − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 3 log ( 3 ) − − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 6 log ( 2 ) + − 5 log ( 3 ) log ( 5 ) − 5 log ( 2 ) log ( 5 ) + 5 log ( 2 ) log ( 3 ) + 5 log ( 5 ) 2 6 log ( 3 ) + − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 2 log ( 2 ) + − log ( 3 ) log ( 5 ) − log ( 2 ) log ( 5 ) + log ( 2 ) log ( 3 ) + log ( 5 ) 2 log ( 5 )
log(5)/(log(5)^2 + log(2)*log(3) - log(2)*log(5) - log(3)*log(5)) - 6*log(2)/(5*log(5)^2 - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3)) - 3*log(3)/(log(5)^2 + log(2)*log(3) - log(2)*log(5) - log(3)*log(5)) + 2*log(2)/(log(5)^2 + log(2)*log(3) - log(2)*log(5) - log(3)*log(5)) + 6*log(3)/(5*log(5)^2 - 5*log(2)*log(5) - 5*log(3)*log(5) + 5*log(2)*log(3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.
