Sr Examen
Integral de a^2/(1+(x^(3/2)*a^2)) dx
Solución
Ha introducido
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oo / | | 2 | a | ----------- dx | 3/2 2 | 1 + x *a | / 1
$$\int\limits_{1}^{\infty} \frac{a^{2}}{a^{2} x^{\frac{3}{2}} + 1}\, dx$$
Integral(a^2/(1 + x^(3/2)*a^2), (x, 1, oo))
Solución detallada
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La integral del producto de una función por una constante es la constante por la integral de esta función:
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No puedo encontrar los pasos en la búsqueda de esta integral.
Pero la integral
Por lo tanto, el resultado es:
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Ahora simplificar:
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Añadimos la constante de integración:
Respuesta:
Respuesta (Indefinida)
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// / ___ 2/3 ___ ___\ \
|| 2/3 ___ | \/ 3 2*(-1) *\/ 3 *\/ x | |
|| 2*(-1) *\/ 3 *atan|- ----- + ---------------------| |
|| / ____\ / 2/3 ____\ | 3 ____ | |
/ || 2/3 | ___ 3 ____ / 1 | 2/3 | 2/3 /1 \ 3 ____ ___ / 1 | | / 1 | |
| || 2*(-1) *log|\/ x - \/ -1 * / -- | (-1) *log|4*x + 4*(-1) *|--| + 4*\/ -1 *\/ x * / -- | | 3* / -- | |
| 2 || | 3 / 2 | | | 2| 3 / 2 | | 3 / 2 | |
| a 2 || \ \/ a / \ \a / \/ a / \ \/ a / |
| ----------- dx = C + a *|<- --------------------------------------- + --------------------------------------------------------------- + ----------------------------------------------------- for a != 0|
| 3/2 2 || ____ ____ ____ |
| 1 + x *a || 2 / 1 2 / 1 2 / 1 |
| || 3*a * / -- 3*a * / -- 3*a * / -- |
/ || 3 / 2 3 / 2 3 / 2 |
|| \/ a \/ a \/ a |
|| |
|| x otherwise |
\\ /
$$\int \frac{a^{2}}{a^{2} x^{\frac{3}{2}} + 1}\, dx = C + a^{2} \left(\begin{cases} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(\sqrt{x} - \sqrt[3]{-1} \sqrt[3]{\frac{1}{a^{2}}} \right)}}{3 a^{2} \sqrt[3]{\frac{1}{a^{2}}}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \sqrt[3]{-1} \sqrt{x} \sqrt[3]{\frac{1}{a^{2}}} + 4 x + 4 \left(-1\right)^{\frac{2}{3}} \left(\frac{1}{a^{2}}\right)^{\frac{2}{3}} \right)}}{3 a^{2} \sqrt[3]{\frac{1}{a^{2}}}} + \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \sqrt{x}}{3 \sqrt[3]{\frac{1}{a^{2}}}} - \frac{\sqrt{3}}{3} \right)}}{3 a^{2} \sqrt[3]{\frac{1}{a^{2}}}} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ / / / 2*pi*I\ / pi*I\ / 5*pi*I\ / 4*pi*I\\ / / 4*pi*I\ / 2*pi*I\\ \ | | | -2*pi*I | ------| -pi*I | ----| pi*I | ------| 2*pi*I | ------|| | -2*pi*I | ------| 2*pi*I | ------|| | | | | / pi*I\ ------- | 3 | ------ | 3 | ---- | 3 | ------ | 3 || | ------- | 3 | ------ | 3 || | | | | 2/3 / 1 \ 2/3 | e | 2/3 3 | e | 2/3 3 | e | 2/3 3 | e | 2/3 3 | e || | 8/3 / 1 \ 8/3 3 | e | 8/3 3 | e || | | | | a *log|1 - ----| a *log|1 - -----| a *e *log|1 - -------| a *e *log|1 - -----| a *e *log|1 - -------| a *e *log|1 - -------|| | 2*a *log|1 - ----| 2*a *e *log|1 - -------| 2*a *e *log|1 - -------|| | | | | | 2/3| | 2/3| | 2/3 | | 2/3| | 2/3 | | 2/3 || | | 4/3| | 4/3 | | 4/3 || | | | | \ a / \ a / \ a / \ a / \ a / \ a /| | \ a / \ a / \ a /| | | |pi*|- ------------------ + ------------------- - ------------------------------ - --------------------------- - --------------------------- - -----------------------------|*Gamma(1/6) pi*|- -------------------- - -------------------------------- - -------------------------------|*Gamma(2/3)| | 2/3 | \ 6 6 6 6 6 6 / \ 3 3 3 / | |a *|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------| | | 2/3 8/3 | | \ a *Gamma(7/6) a *Gamma(5/3) / |------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ for 4*|arg(a)| < 2*pi < 3*pi | | oo | / | | | | 2 | | a | | ----------- dx otherwise | | 2 3/2 | | 1 + a *x | | | / | 1 \
$$\begin{cases} \frac{a^{\frac{2}{3}} \left(\frac{\pi \left(- \frac{a^{\frac{2}{3}} \log{\left(1 - \frac{1}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{e^{\frac{i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{2 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} + \frac{a^{\frac{2}{3}} \log{\left(1 - \frac{e^{i \pi}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{4 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(1 - \frac{e^{\frac{5 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6}\right) \Gamma\left(\frac{1}{6}\right)}{a^{\frac{2}{3}} \Gamma\left(\frac{7}{6}\right)} - \frac{\pi \left(- \frac{2 a^{\frac{8}{3}} \log{\left(1 - \frac{1}{a^{\frac{4}{3}}} \right)}}{3} - \frac{2 a^{\frac{8}{3}} e^{\frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{2 i \pi}{3}}}{a^{\frac{4}{3}}} \right)}}{3} - \frac{2 a^{\frac{8}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{4 i \pi}{3}}}{a^{\frac{4}{3}}} \right)}}{3}\right) \Gamma\left(\frac{2}{3}\right)}{a^{\frac{8}{3}} \Gamma\left(\frac{5}{3}\right)}\right)}{3 \pi} & \text{for}\: 4 \left|{\arg{\left(a \right)}}\right| < 2 \pi \\\int\limits_{1}^{\infty} \frac{a^{2}}{a^{2} x^{\frac{3}{2}} + 1}\, dx & \text{otherwise} \end{cases}$$
=
=
/ / / / 2*pi*I\ / pi*I\ / 5*pi*I\ / 4*pi*I\\ / / 4*pi*I\ / 2*pi*I\\ \ | | | -2*pi*I | ------| -pi*I | ----| pi*I | ------| 2*pi*I | ------|| | -2*pi*I | ------| 2*pi*I | ------|| | | | | / pi*I\ ------- | 3 | ------ | 3 | ---- | 3 | ------ | 3 || | ------- | 3 | ------ | 3 || | | | | 2/3 / 1 \ 2/3 | e | 2/3 3 | e | 2/3 3 | e | 2/3 3 | e | 2/3 3 | e || | 8/3 / 1 \ 8/3 3 | e | 8/3 3 | e || | | | | a *log|1 - ----| a *log|1 - -----| a *e *log|1 - -------| a *e *log|1 - -----| a *e *log|1 - -------| a *e *log|1 - -------|| | 2*a *log|1 - ----| 2*a *e *log|1 - -------| 2*a *e *log|1 - -------|| | | | | | 2/3| | 2/3| | 2/3 | | 2/3| | 2/3 | | 2/3 || | | 4/3| | 4/3 | | 4/3 || | | | | \ a / \ a / \ a / \ a / \ a / \ a /| | \ a / \ a / \ a /| | | |pi*|- ------------------ + ------------------- - ------------------------------ - --------------------------- - --------------------------- - -----------------------------|*Gamma(1/6) pi*|- -------------------- - -------------------------------- - -------------------------------|*Gamma(2/3)| | 2/3 | \ 6 6 6 6 6 6 / \ 3 3 3 / | |a *|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------| | | 2/3 8/3 | | \ a *Gamma(7/6) a *Gamma(5/3) / |------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ for 4*|arg(a)| < 2*pi < 3*pi | | oo | / | | | | 2 | | a | | ----------- dx otherwise | | 2 3/2 | | 1 + a *x | | | / | 1 \
$$\begin{cases} \frac{a^{\frac{2}{3}} \left(\frac{\pi \left(- \frac{a^{\frac{2}{3}} \log{\left(1 - \frac{1}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \log{\left(1 - \frac{e^{\frac{i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{2 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} + \frac{a^{\frac{2}{3}} \log{\left(1 - \frac{e^{i \pi}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{4 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6} - \frac{a^{\frac{2}{3}} e^{\frac{i \pi}{3}} \log{\left(1 - \frac{e^{\frac{5 i \pi}{3}}}{a^{\frac{2}{3}}} \right)}}{6}\right) \Gamma\left(\frac{1}{6}\right)}{a^{\frac{2}{3}} \Gamma\left(\frac{7}{6}\right)} - \frac{\pi \left(- \frac{2 a^{\frac{8}{3}} \log{\left(1 - \frac{1}{a^{\frac{4}{3}}} \right)}}{3} - \frac{2 a^{\frac{8}{3}} e^{\frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{2 i \pi}{3}}}{a^{\frac{4}{3}}} \right)}}{3} - \frac{2 a^{\frac{8}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(1 - \frac{e^{\frac{4 i \pi}{3}}}{a^{\frac{4}{3}}} \right)}}{3}\right) \Gamma\left(\frac{2}{3}\right)}{a^{\frac{8}{3}} \Gamma\left(\frac{5}{3}\right)}\right)}{3 \pi} & \text{for}\: 4 \left|{\arg{\left(a \right)}}\right| < 2 \pi \\\int\limits_{1}^{\infty} \frac{a^{2}}{a^{2} x^{\frac{3}{2}} + 1}\, dx & \text{otherwise} \end{cases}$$
Piecewise((a^(2/3)*(pi*(-a^(2/3)*log(1 - 1/a^(2/3))/6 + a^(2/3)*log(1 - exp_polar(pi*i)/a^(2/3))/6 - a^(2/3)*exp(-2*pi*i/3)*log(1 - exp_polar(2*pi*i/3)/a^(2/3))/6 - a^(2/3)*exp(-pi*i/3)*log(1 - exp_polar(pi*i/3)/a^(2/3))/6 - a^(2/3)*exp(pi*i/3)*log(1 - exp_polar(5*pi*i/3)/a^(2/3))/6 - a^(2/3)*exp(2*pi*i/3)*log(1 - exp_polar(4*pi*i/3)/a^(2/3))/6)*gamma(1/6)/(a^(2/3)*gamma(7/6)) - pi*(-2*a^(8/3)*log(1 - 1/a^(4/3))/3 - 2*a^(8/3)*exp(-2*pi*i/3)*log(1 - exp_polar(4*pi*i/3)/a^(4/3))/3 - 2*a^(8/3)*exp(2*pi*i/3)*log(1 - exp_polar(2*pi*i/3)/a^(4/3))/3)*gamma(2/3)/(a^(8/3)*gamma(5/3)))/(3*pi), 4*Abs(arg(a)) < 2*pi), (Integral(a^2/(1 + a^2*x^(3/2)), (x, 1, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.