Sr Examen
Integral de log(1+x)/x^n dx
Solución
Ha introducido
[src]
oo / | | log(1 + x) | ---------- dx | n | x | / 0
$$\int\limits_{0}^{\infty} \frac{\log{\left(x + 1 \right)}}{x^{n}}\, dx$$
Integral(log(1 + x)/x^n, (x, 0, oo))
Solución detallada
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Usamos la integración por partes:
que y que .
Entonces .
Para buscar :
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Integral es when :
Ahora resolvemos podintegral.
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
[src]
// 2 -n / pi*I \ 2 -n / pi*I \ \
|| 2*x *x *Gamma(2 - n)*lerchphi\x*e , 1, 2 - n/ n*x *x *Gamma(2 - n)*lerchphi\x*e , 1, 2 - n/ |
|| ------------------------------------------------- - ------------------------------------------------- |
|| Gamma(3 - n) Gamma(3 - n) |
|| ----------------------------------------------------------------------------------------------------- for And(n > -oo, n < oo, n != 1)|
/ || 1 - n | // 1 - n \
| || | ||x |
| log(1 + x) ||/ -polylog(2, 1 + x) + pi*I*log(1 + x) for |1 + x| < 1 | ||------ for n != 1|
| ---------- dx = C - |<| | + |<1 - n |*log(1 + x)
| n ||| / 1 \ 1 | || |
| x ||| -polylog(2, 1 + x) - pi*I*log|-----| for ------- < 1 | ||log(x) otherwise |
| ||< \1 + x/ |1 + x| otherwise | \\ /
/ ||| |
||| __0, 2 /1, 1 | \ __2, 0 / 1, 1 | \ |
|||-polylog(2, 1 + x) + pi*I*/__ | | 1 + x| - pi*I*/__ | | 1 + x| otherwise |
||\ \_|2, 2 \ 0, 0 | / \_|2, 2 \0, 0 | / |
\\ /
$$\int \frac{\log{\left(x + 1 \right)}}{x^{n}}\, dx = C + \left(\begin{cases} \frac{x^{1 - n}}{1 - n} & \text{for}\: n \neq 1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}\right) \log{\left(x + 1 \right)} - \begin{cases} \frac{- \frac{n x^{2} x^{- n} \Phi\left(x e^{i \pi}, 1, 2 - n\right) \Gamma\left(2 - n\right)}{\Gamma\left(3 - n\right)} + \frac{2 x^{2} x^{- n} \Phi\left(x e^{i \pi}, 1, 2 - n\right) \Gamma\left(2 - n\right)}{\Gamma\left(3 - n\right)}}{1 - n} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 1 \\\begin{cases} i \pi \log{\left(x + 1 \right)} - \operatorname{Li}_{2}\left(x + 1\right) & \text{for}\: \left|{x + 1}\right| < 1 \\- i \pi \log{\left(\frac{1}{x + 1} \right)} - \operatorname{Li}_{2}\left(x + 1\right) & \text{for}\: \frac{1}{\left|{x + 1}\right|} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left(\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle| {x + 1} \right)} + i \pi {G_{2, 2}^{0, 2}\left(\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle| {x + 1} \right)} - \operatorname{Li}_{2}\left(x + 1\right) & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ -pi | ------------------ for And(re(n) > 1, -1 + re(n) < 1) | (-1 + n)*sin(pi*n) | | oo | / < | | | -n | | x *log(1 + x) dx otherwise | | |/ |0 \
$$\begin{cases} - \frac{\pi}{\left(n - 1\right) \sin{\left(\pi n \right)}} & \text{for}\: \operatorname{re}{\left(n\right)} > 1 \wedge \operatorname{re}{\left(n\right)} - 1 < 1 \\\int\limits_{0}^{\infty} x^{- n} \log{\left(x + 1 \right)}\, dx & \text{otherwise} \end{cases}$$
=
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/ -pi | ------------------ for And(re(n) > 1, -1 + re(n) < 1) | (-1 + n)*sin(pi*n) | | oo | / < | | | -n | | x *log(1 + x) dx otherwise | | |/ |0 \
$$\begin{cases} - \frac{\pi}{\left(n - 1\right) \sin{\left(\pi n \right)}} & \text{for}\: \operatorname{re}{\left(n\right)} > 1 \wedge \operatorname{re}{\left(n\right)} - 1 < 1 \\\int\limits_{0}^{\infty} x^{- n} \log{\left(x + 1 \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-pi/((-1 + n)*sin(pi*n)), (re(n) > 1)∧(-1 + re(n) < 1)), (Integral(x^(-n)*log(1 + x), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.