Sr Examen
Integral de 1/((1-x)^(1/3)*(2-x)^2) dx
Solución
Ha introducido
[src]
oo / | | 1 | ------------------ dx | 3 _______ 2 | \/ 1 - x *(2 - x) | / 4
$$\int\limits_{4}^{\infty} \frac{1}{\sqrt[3]{1 - x} \left(2 - x\right)^{2}}\, dx$$
Integral(1/((1 - x)^(1/3)*(2 - x)^2), (x, 4, oo))
Respuesta (Indefinida)
[src]
/ 4*pi*I\ / 4*pi*I\ 2*pi*I -2*pi*I / 2*pi*I\ 2*pi*I -2*pi*I / 2*pi*I\ 2*pi*I / | ------| | ------| ------ ------- | ------| ------ ------- | ------| ------ | 2/3 4/3 | 3 ________ 3 | 2/3 7/3 | 3 ________ 3 | 2/3 2 3 2/3 4/3 3 | 3 ________ 3 | 2/3 4/3 3 / 3 ________ 2*pi*I\ 2/3 7/3 3 | 3 ________ 3 | 2/3 7/3 3 / 3 ________ 2*pi*I\ | 1 2*(-1) *(-1 + x) *Gamma(2/3)*log\1 - \/ -1 + x *e / 2*(-1) *(-1 + x) *Gamma(2/3)*log\1 - \/ -1 + x *e / 6*(-1) *(-1 + x) *e *Gamma(2/3) 2*(-1) *(-1 + x) *e *Gamma(2/3)*log\1 - \/ -1 + x *e / 2*(-1) *(-1 + x) *e *Gamma(2/3)*log\1 - \/ -1 + x *e / 2*(-1) *(-1 + x) *e *Gamma(2/3)*log\1 - \/ -1 + x *e / 2*(-1) *(-1 + x) *e *Gamma(2/3)*log\1 - \/ -1 + x *e / | ------------------ dx = C - --------------------------------------------------------------------- + --------------------------------------------------------------------- + --------------------------------------------------------------------- - --------------------------------------------------------------------- - --------------------------------------------------------------------- + --------------------------------------------------------------------- + --------------------------------------------------------------------- | 3 _______ 2 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I 2*pi*I | \/ 1 - x *(2 - x) ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ | 4/3 3 7/3 3 4/3 3 7/3 3 4/3 3 7/3 3 4/3 3 7/3 3 4/3 3 7/3 3 4/3 3 7/3 3 4/3 3 7/3 3 / - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3) - 9*(-1 + x) *e *Gamma(5/3) + 9*(-1 + x) *e *Gamma(5/3)
$$\int \frac{1}{\sqrt[3]{1 - x} \left(2 - x\right)^{2}}\, dx = C + \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{7}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(- \sqrt[3]{x - 1} e^{\frac{2 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} + \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{7}{3}} \log{\left(- \sqrt[3]{x - 1} e^{\frac{4 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} + \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \log{\left(- \sqrt[3]{x - 1} e^{2 i \pi} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} - \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{4}{3}} e^{- \frac{2 i \pi}{3}} \log{\left(- \sqrt[3]{x - 1} e^{\frac{2 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} - \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{4}{3}} \log{\left(- \sqrt[3]{x - 1} e^{\frac{4 i \pi}{3}} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} - \frac{2 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \log{\left(- \sqrt[3]{x - 1} e^{2 i \pi} + 1 \right)} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)} + \frac{6 \left(-1\right)^{\frac{2}{3}} \left(x - 1\right)^{2} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right)}{9 \left(x - 1\right)^{\frac{7}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right) - 9 \left(x - 1\right)^{\frac{4}{3}} e^{\frac{2 i \pi}{3}} \Gamma\left(\frac{5}{3}\right)}$$
Respuesta
[src]
-pi*I
------ _
2/3 3 |_ /1/3, 4/3 | \
2 *e *Gamma(4/3)* | | | -1/2|
2 1 \ 7/3 | /
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4*Gamma(7/3)
$$\frac{2^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \Gamma\left(\frac{4}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle| {- \frac{1}{2}} \right)}}{4 \Gamma\left(\frac{7}{3}\right)}$$
=
=
-pi*I
------ _
2/3 3 |_ /1/3, 4/3 | \
2 *e *Gamma(4/3)* | | | -1/2|
2 1 \ 7/3 | /
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4*Gamma(7/3)
$$\frac{2^{\frac{2}{3}} e^{- \frac{i \pi}{3}} \Gamma\left(\frac{4}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle| {- \frac{1}{2}} \right)}}{4 \Gamma\left(\frac{7}{3}\right)}$$
2^(2/3)*exp(-pi*i/3)*gamma(4/3)*hyper((1/3, 4/3), (7/3,), -1/2)/(4*gamma(7/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.