Sr Examen
Integral de sqr(x)*cbrt((3-x^3)^2) dx
Solución
Ha introducido
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___ \/ 3 / | | 2 | ___________ | / 2 | 3 / / 3\ | x*\/ \3 - x / dx | / 1
$$\int\limits_{1}^{\sqrt{3}} x \left(\sqrt[3]{\left(3 - x^{3}\right)^{2}}\right)^{2}\, dx$$
Integral(x*(((3 - x^3)^2)^(1/3))^2, (x, 1, sqrt(3)))
Respuesta
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___ \/ 3 / | | / -2*pi*I -2*pi*I | | ------- _ / | 3\ ------- _ / | 3\ | | 3 ___ 3 |_ |-4/3, 2/3 | x | 3 ___ 4 3 |_ |-1/3, 5/3 | x | | | 2*x*\/ 3 *e *Gamma(2/3)* | | | --| 8*\/ 3 *x *e *Gamma(2/3)* | | | --| | | 2 1 \ 5/3 | 3 / 2 1 \ 8/3 | 3 / 3 | | --------------------------------------------------- - ---------------------------------------------------- for -3 + x >= 0 | | Gamma(5/3) 15*Gamma(5/3) | < dx | | | | _ / | 3 2*pi*I\ _ / | 3 2*pi*I\ | | 3 ___ |_ |-4/3, 2/3 | x *e | 3 ___ 4 2*pi*I |_ |-1/3, 5/3 | x *e | | |2*x*\/ 3 *Gamma(2/3)* | | | ----------| 8*\/ 3 *x *e *Gamma(2/3)* | | | ----------| | | 2 1 \ 5/3 | 3 / 2 1 \ 8/3 | 3 / | |-------------------------------------------------- - ----------------------------------------------------------- otherwise | \ Gamma(5/3) 15*Gamma(5/3) | / 1
$$\int\limits_{1}^{\sqrt{3}} \begin{cases} - \frac{8 \sqrt[3]{3} x^{4} e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{for}\: x^{3} - 3 \geq 0 \\- \frac{8 \sqrt[3]{3} x^{4} e^{2 i \pi} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx$$
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___ \/ 3 / | | / -2*pi*I -2*pi*I | | ------- _ / | 3\ ------- _ / | 3\ | | 3 ___ 3 |_ |-4/3, 2/3 | x | 3 ___ 4 3 |_ |-1/3, 5/3 | x | | | 2*x*\/ 3 *e *Gamma(2/3)* | | | --| 8*\/ 3 *x *e *Gamma(2/3)* | | | --| | | 2 1 \ 5/3 | 3 / 2 1 \ 8/3 | 3 / 3 | | --------------------------------------------------- - ---------------------------------------------------- for -3 + x >= 0 | | Gamma(5/3) 15*Gamma(5/3) | < dx | | | | _ / | 3 2*pi*I\ _ / | 3 2*pi*I\ | | 3 ___ |_ |-4/3, 2/3 | x *e | 3 ___ 4 2*pi*I |_ |-1/3, 5/3 | x *e | | |2*x*\/ 3 *Gamma(2/3)* | | | ----------| 8*\/ 3 *x *e *Gamma(2/3)* | | | ----------| | | 2 1 \ 5/3 | 3 / 2 1 \ 8/3 | 3 / | |-------------------------------------------------- - ----------------------------------------------------------- otherwise | \ Gamma(5/3) 15*Gamma(5/3) | / 1
$$\int\limits_{1}^{\sqrt{3}} \begin{cases} - \frac{8 \sqrt[3]{3} x^{4} e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x e^{- \frac{2 i \pi}{3}} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{for}\: x^{3} - 3 \geq 0 \\- \frac{8 \sqrt[3]{3} x^{4} e^{2 i \pi} \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{15 \Gamma\left(\frac{5}{3}\right)} + \frac{2 \sqrt[3]{3} x \Gamma\left(\frac{2}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle| {\frac{x^{3} e^{2 i \pi}}{3}} \right)}}{\Gamma\left(\frac{5}{3}\right)} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((2*x*3^(1/3)*exp(-2*pi*i/3)*gamma(2/3)*hyper((-4/3, 2/3), (5/3,), x^3/3)/gamma(5/3) - 8*3^(1/3)*x^4*exp(-2*pi*i/3)*gamma(2/3)*hyper((-1/3, 5/3), (8/3,), x^3/3)/(15*gamma(5/3)), -3 + x^3 >= 0), (2*x*3^(1/3)*gamma(2/3)*hyper((-4/3, 2/3), (5/3,), x^3*exp_polar(2*pi*i)/3)/gamma(5/3) - 8*3^(1/3)*x^4*exp_polar(2*pi*i)*gamma(2/3)*hyper((-1/3, 5/3), (8/3,), x^3*exp_polar(2*pi*i)/3)/(15*gamma(5/3)), True)), (x, 1, sqrt(3)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.