Sr Examen
Integral de 1ln((1+sqrtx))/(x^1/3) dx
Solución
Ha introducido
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3 / | | / ___\ | log\1 + \/ x / | -------------- dx | 3 ___ | \/ x | / 1
$$\int\limits_{1}^{3} \frac{\log{\left(\sqrt{x} + 1 \right)}}{\sqrt[3]{x}}\, dx$$
Integral(log(1 + sqrt(x))/x^(1/3), (x, 1, 3))
Respuesta (Indefinida)
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/ ___ / 1 6 ___\\ / |2*\/ 3 *|- - + \/ x || | ___ | \ 2 /| | / ___\ 2/3 / 6 ___\ / 3 ___ 6 ___\ 6 ___ 3*\/ 3 *atan|---------------------| 2/3 / ___\ | log\1 + \/ x / 9*x 3*log\1 + \/ x / 3*log\1 + \/ x - \/ x / 9*\/ x \ 3 / 3*x *log\1 + \/ x / | -------------- dx = C - ------ - ---------------- + ------------------------ + ------- - ----------------------------------- + --------------------- | 3 ___ 8 2 4 2 2 2 | \/ x | /
$$\int \frac{\log{\left(\sqrt{x} + 1 \right)}}{\sqrt[3]{x}}\, dx = C + \frac{9 \sqrt[6]{x}}{2} + \frac{3 x^{\frac{2}{3}} \log{\left(\sqrt{x} + 1 \right)}}{2} - \frac{9 x^{\frac{2}{3}}}{8} - \frac{3 \log{\left(\sqrt[6]{x} + 1 \right)}}{2} + \frac{3 \log{\left(- \sqrt[6]{x} + \sqrt[3]{x} + 1 \right)}}{4} - \frac{3 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(\sqrt[6]{x} - \frac{1}{2}\right)}{3} \right)}}{2}$$
Respuesta
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-pi*I / pi*I\ pi*I / 5*pi*I\ -pi*I / pi*I\ pi*I / 5*pi*I\
------ | ----| ---- | ------| ------ | ----| ---- | ------|
2/3 / 6 ___ pi*I\ 6 ___ 3 | 3 | 3 | 3 | 2/3 / ___ pi*I\ 3 | 6 ___ 3 | 3 | 6 ___ 3 |
63*Gamma(7/3) 21*3 *Gamma(7/3) 7*Gamma(7/3)*log\1 - \/ 3 *e / 21*\/ 3 *Gamma(7/3) 7*e *Gamma(7/3)*log\1 - e / 7*e *Gamma(7/3)*log\1 - e / 7*3 *Gamma(7/3)*log\1 - \/ 3 *e / 7*e *Gamma(7/3)*log\1 - \/ 3 *e / 7*e *Gamma(7/3)*log\1 - \/ 3 *e /
- ------------- - ------------------ - --------------------------------- + ------------------- - ----------------------------------- - ----------------------------------- + -------------------------------------- + ----------------------------------------- + -----------------------------------------
8*Gamma(10/3) 8*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3)
$$- \frac{7 e^{\frac{i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 e^{\frac{i \pi}{3}} \log{\left(- \sqrt[6]{3} e^{\frac{5 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} - \frac{63 \Gamma\left(\frac{7}{3}\right)}{8 \Gamma\left(\frac{10}{3}\right)} - \frac{21 \cdot 3^{\frac{2}{3}} \Gamma\left(\frac{7}{3}\right)}{8 \Gamma\left(\frac{10}{3}\right)} - \frac{7 \log{\left(1 - \sqrt[6]{3} e^{i \pi} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 \cdot 3^{\frac{2}{3}} \log{\left(1 - \sqrt{3} e^{i \pi} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{21 \sqrt[6]{3} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 e^{- \frac{i \pi}{3}} \log{\left(1 - \sqrt[6]{3} e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} - \frac{7 e^{- \frac{i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)}$$
=
=
-pi*I / pi*I\ pi*I / 5*pi*I\ -pi*I / pi*I\ pi*I / 5*pi*I\
------ | ----| ---- | ------| ------ | ----| ---- | ------|
2/3 / 6 ___ pi*I\ 6 ___ 3 | 3 | 3 | 3 | 2/3 / ___ pi*I\ 3 | 6 ___ 3 | 3 | 6 ___ 3 |
63*Gamma(7/3) 21*3 *Gamma(7/3) 7*Gamma(7/3)*log\1 - \/ 3 *e / 21*\/ 3 *Gamma(7/3) 7*e *Gamma(7/3)*log\1 - e / 7*e *Gamma(7/3)*log\1 - e / 7*3 *Gamma(7/3)*log\1 - \/ 3 *e / 7*e *Gamma(7/3)*log\1 - \/ 3 *e / 7*e *Gamma(7/3)*log\1 - \/ 3 *e /
- ------------- - ------------------ - --------------------------------- + ------------------- - ----------------------------------- - ----------------------------------- + -------------------------------------- + ----------------------------------------- + -----------------------------------------
8*Gamma(10/3) 8*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3) 2*Gamma(10/3)
$$- \frac{7 e^{\frac{i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 e^{\frac{i \pi}{3}} \log{\left(- \sqrt[6]{3} e^{\frac{5 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} - \frac{63 \Gamma\left(\frac{7}{3}\right)}{8 \Gamma\left(\frac{10}{3}\right)} - \frac{21 \cdot 3^{\frac{2}{3}} \Gamma\left(\frac{7}{3}\right)}{8 \Gamma\left(\frac{10}{3}\right)} - \frac{7 \log{\left(1 - \sqrt[6]{3} e^{i \pi} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 \cdot 3^{\frac{2}{3}} \log{\left(1 - \sqrt{3} e^{i \pi} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{21 \sqrt[6]{3} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} + \frac{7 e^{- \frac{i \pi}{3}} \log{\left(1 - \sqrt[6]{3} e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)} - \frac{7 e^{- \frac{i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{7}{3}\right)}{2 \Gamma\left(\frac{10}{3}\right)}$$
-63*gamma(7/3)/(8*gamma(10/3)) - 21*3^(2/3)*gamma(7/3)/(8*gamma(10/3)) - 7*gamma(7/3)*log(1 - 3^(1/6)*exp_polar(pi*i))/(2*gamma(10/3)) + 21*3^(1/6)*gamma(7/3)/(2*gamma(10/3)) - 7*exp(-pi*i/3)*gamma(7/3)*log(1 - exp_polar(pi*i/3))/(2*gamma(10/3)) - 7*exp(pi*i/3)*gamma(7/3)*log(1 - exp_polar(5*pi*i/3))/(2*gamma(10/3)) + 7*3^(2/3)*gamma(7/3)*log(1 - sqrt(3)*exp_polar(pi*i))/(2*gamma(10/3)) + 7*exp(-pi*i/3)*gamma(7/3)*log(1 - 3^(1/6)*exp_polar(pi*i/3))/(2*gamma(10/3)) + 7*exp(pi*i/3)*gamma(7/3)*log(1 - 3^(1/6)*exp_polar(5*pi*i/3))/(2*gamma(10/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.