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Expresiones idénticas
lnx/e^(x+lnx)
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Expresiones semejantes
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Expresiones con funciones
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lnx/(x²+1)
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lnx/2

Resolución de integrales/ x+lnx/ lnx/e^(x+lnx)

Integral de lnx/e^(x+lnx) dx

Solución

Ha introducido [src]

  2               
  /               
 |                
 |     log(x)     
 |  ----------- dx
 |   x + log(x)   
 |  E             
 |                
/                 
0

$$\int\limits_{0}^{2} \frac{\log{\left(x \right)}}{e^{x + \log{\left(x \right)}}}\, dx$$

Integral(log(x)/E^(x + log(x)), (x, 0, 2))

Solución detallada

Usamos la integración por partes:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

que $u{\left(x \right)} = \log{\left(x \right)}$ y que $\operatorname{dv}{\left(x \right)} = \frac{e^{- x}}{x}$ .

Entonces $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .

Para buscar $v{\left(x \right)}$ :
Ahora resolvemos podintegral.
No puedo encontrar los pasos en la búsqueda de esta integral.

Pero la integral

$\begin{cases} - x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + i \pi \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\- x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi \log{\left(\frac{1}{x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + 2 i \pi \log{\left(x \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\- x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} - i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + 2 i \pi \log{\left(x \right)} & \text{otherwese} \end{cases}$
Ahora simplificar:

$\begin{cases} x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - i \pi \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - i \pi \log{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi \left(\begin{cases} 0 & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(\frac{1}{x} \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - i \pi \left(\begin{cases} - \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\0 & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{otherwese} \end{cases}$
Añadimos la constante de integración:

$\begin{cases} x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - i \pi \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - i \pi \log{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi \left(\begin{cases} 0 & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(\frac{1}{x} \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - i \pi \left(\begin{cases} - \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\0 & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - i \pi \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} - i \pi \log{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi \left(\begin{cases} 0 & \text{for}\: \left|{x}\right| < 1 \\- \log{\left(\frac{1}{x} \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - i \pi \left(\begin{cases} - \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\0 & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\{G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} & \text{otherwese} \end{cases}\right) - \frac{\log{\left(x \right)}^{2}}{2} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)} - \gamma \log{\left(x \right)} - 2 i \pi \log{\left(x \right)} & \text{otherwese} \end{cases}+ \mathrm{constant}$

Respuesta (Indefinida) [src]

                        //                                    2                              _                                                                            \                
                        ||                                 log (x)                          |_  /1, 1, 1 |   \                                                            |                
                        ||                                 ------- + EulerGamma*log(x) - x* |   |        | -x| + pi*I*log(x)                                   for |x| < 1|                
                        ||                                    2                            3  3 \2, 2, 2 |   /                                                            |                
  /                     ||                                                                                                                                                |                
 |                      ||                            2                              _                                                                                    |                
 |    log(x)            ||                         log (x)                          |_  /1, 1, 1 |   \           /1\                                                1     |                
 | ----------- dx = C - |<                         ------- + EulerGamma*log(x) - x* |   |        | -x| + pi*I*log|-| + 2*pi*I*log(x)                           for --- < 1| + Ei(-x)*log(x)
 |  x + log(x)          ||                            2                            3  3 \2, 2, 2 |   /           \x/                                               |x|    |                
 | E                    ||                                                                                                                                                |                
 |                      ||   2                              _                                                                                                             |                
/                       ||log (x)                          |_  /1, 1, 1 |   \         __2, 0 /      1, 1 |  \         __0, 2 /1, 1       |  \                             |                
                        ||------- + EulerGamma*log(x) - x* |   |        | -x| + pi*I*/__     |           | x| - pi*I*/__     |           | x| + 2*pi*I*log(x)   otherwise |                
                        ||   2                            3  3 \2, 2, 2 |   /        \_|2, 2 \0, 0       |  /        \_|2, 2 \      0, 0 |  /                             |                
                        \\                                                                                                                                                /

$$\int \frac{\log{\left(x \right)}}{e^{x + \log{\left(x \right)}}}\, dx = C - \begin{cases} - x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + i \pi \log{\left(x \right)} & \text{for}\: \left|{x}\right| < 1 \\- x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi \log{\left(\frac{1}{x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + 2 i \pi \log{\left(x \right)} & \text{for}\: \frac{1}{\left|{x}\right|} < 1 \\- x {{}_{3}F_{3}\left(\begin{matrix} 1, 1, 1 \\ 2, 2, 2 \end{matrix}\middle| {- x} \right)} + i \pi {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x} \right)} - i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x} \right)} + \frac{\log{\left(x \right)}^{2}}{2} + \gamma \log{\left(x \right)} + 2 i \pi \log{\left(x \right)} & \text{otherwise} \end{cases} + \log{\left(x \right)} \operatorname{Ei}{\left(- x \right)}$$

Simplificar

Respuesta [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Respuesta numérica [src]

-940.702025586466

-940.702025586466

Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

Integral de lnx/e^(x+lnx) dx

Límites de integración:

Gráfico:

Definida a trozos:

Solución