Sr Examen
Integral de (x^4dx)/(4+x^6)^(1/2) dx
Solución
Ha introducido
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1 / | | 4 | x | ----------- dx | ________ | / 6 | \/ 4 + x | / 0
$$\int\limits_{0}^{1} \frac{x^{4}}{\sqrt{x^{6} + 4}}\, dx$$
Integral(x^4/sqrt(4 + x^6), (x, 0, 1))
Respuesta (Indefinida)
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/ _ / | 6 pi*I\ | 5 |_ |1/2, 5/6 | x *e | | 4 x *Gamma(5/6)* | | | --------| | x 2 1 \ 11/6 | 4 / | ----------- dx = C + ---------------------------------------- | ________ 12*Gamma(11/6) | / 6 | \/ 4 + x | /
$$\int \frac{x^{4}}{\sqrt{x^{6} + 4}}\, dx = C + \frac{x^{5} \Gamma\left(\frac{5}{6}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle| {\frac{x^{6} e^{i \pi}}{4}} \right)}}{12 \Gamma\left(\frac{11}{6}\right)}$$
Gráfica
Respuesta
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_
|_ /1/2, 5/6 | \
Gamma(5/6)* | | | -1/4|
2 1 \ 11/6 | /
---------------------------------
12*Gamma(11/6)
$$\frac{\Gamma\left(\frac{5}{6}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{12 \Gamma\left(\frac{11}{6}\right)}$$
=
=
_
|_ /1/2, 5/6 | \
Gamma(5/6)* | | | -1/4|
2 1 \ 11/6 | /
---------------------------------
12*Gamma(11/6)
$$\frac{\Gamma\left(\frac{5}{6}\right) {{}_{2}F_{1}\left(\begin{matrix} \frac{1}{2}, \frac{5}{6} \\ \frac{11}{6} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{12 \Gamma\left(\frac{11}{6}\right)}$$
gamma(5/6)*hyper((1/2, 5/6), (11/6,), -1/4)/(12*gamma(11/6))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.