Sr Examen
Integral de x^(1÷3)/(1+sqrt(x)) dx
Solución
Ha introducido
[src]
1 / | | 3 ___ | \/ x | --------- dx | ___ | 1 + \/ x | / 0
$$\int\limits_{0}^{1} \frac{\sqrt[3]{x}}{\sqrt{x} + 1}\, dx$$
Integral(x^(1/3)/(1 + sqrt(x)), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ -2*pi*I / pi*I\ 2*pi*I / 5*pi*I\ | ------- | ----| ------ | ------| | 3 ___ 3 ___ / 6 ___ pi*I\ 5/6 3 | 6 ___ 3 | 3 | 6 ___ 3 | | \/ x 8*\/ x *Gamma(8/3) 16*Gamma(8/3)*log\1 - \/ x *e / 16*x *Gamma(8/3) 16*e *Gamma(8/3)*log\1 - \/ x *e / 16*e *Gamma(8/3)*log\1 - \/ x *e / | --------- dx = C - ------------------ - ---------------------------------- + ------------------ - ------------------------------------------- - -------------------------------------------- | ___ Gamma(11/3) 3*Gamma(11/3) 5*Gamma(11/3) 3*Gamma(11/3) 3*Gamma(11/3) | 1 + \/ x | /
$$\int \frac{\sqrt[3]{x}}{\sqrt{x} + 1}\, dx = C + \frac{16 x^{\frac{5}{6}} \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{8 \sqrt[3]{x} \Gamma\left(\frac{8}{3}\right)}{\Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(- \sqrt[6]{x} e^{\frac{i \pi}{3}} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(- \sqrt[6]{x} e^{i \pi} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(- \sqrt[6]{x} e^{\frac{5 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)}$$
Respuesta
[src]
-2*pi*I / pi*I\ 2*pi*I / 5*pi*I\
------- | ----| ------ | ------|
/ pi*I\ 3 | 3 | 3 | 3 |
24*Gamma(8/3) 16*Gamma(8/3)*log\1 - e / 16*e *Gamma(8/3)*log\1 - e / 16*e *Gamma(8/3)*log\1 - e /
- ------------- - ---------------------------- - ------------------------------------- - --------------------------------------
5*Gamma(11/3) 3*Gamma(11/3) 3*Gamma(11/3) 3*Gamma(11/3)
$$- \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{24 \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(1 - e^{i \pi} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)}$$
=
=
-2*pi*I / pi*I\ 2*pi*I / 5*pi*I\
------- | ----| ------ | ------|
/ pi*I\ 3 | 3 | 3 | 3 |
24*Gamma(8/3) 16*Gamma(8/3)*log\1 - e / 16*e *Gamma(8/3)*log\1 - e / 16*e *Gamma(8/3)*log\1 - e /
- ------------- - ---------------------------- - ------------------------------------- - --------------------------------------
5*Gamma(11/3) 3*Gamma(11/3) 3*Gamma(11/3) 3*Gamma(11/3)
$$- \frac{16 e^{\frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{5 i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{24 \Gamma\left(\frac{8}{3}\right)}{5 \Gamma\left(\frac{11}{3}\right)} - \frac{16 \log{\left(1 - e^{i \pi} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)} - \frac{16 e^{- \frac{2 i \pi}{3}} \log{\left(1 - e^{\frac{i \pi}{3}} \right)} \Gamma\left(\frac{8}{3}\right)}{3 \Gamma\left(\frac{11}{3}\right)}$$
-24*gamma(8/3)/(5*gamma(11/3)) - 16*gamma(8/3)*log(1 - exp_polar(pi*i))/(3*gamma(11/3)) - 16*exp(-2*pi*i/3)*gamma(8/3)*log(1 - exp_polar(pi*i/3))/(3*gamma(11/3)) - 16*exp(2*pi*i/3)*gamma(8/3)*log(1 - exp_polar(5*pi*i/3))/(3*gamma(11/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.