Sr Examen
Integral de sqrt((x^4)+1)*x^2 dx
Solución
Ha introducido
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1 / | | ________ | / 4 2 | \/ x + 1 *x dx | / 0
$$\int\limits_{0}^{1} x^{2} \sqrt{x^{4} + 1}\, dx$$
Integral(sqrt(x^4 + 1)*x^2, (x, 0, 1))
Respuesta (Indefinida)
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/ _ | 3 |_ /-1/2, 3/4 | 4 pi*I\ | ________ x *Gamma(3/4)* | | | x *e | | / 4 2 2 1 \ 7/4 | / | \/ x + 1 *x dx = C + ----------------------------------------- | 4*Gamma(7/4) /
$$\int x^{2} \sqrt{x^{4} + 1}\, dx = C + \frac{x^{3} \Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {x^{4} e^{i \pi}} \right)}}{4 \Gamma\left(\frac{7}{4}\right)}$$
Gráfica
Respuesta
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_
|_ /-1/2, 3/4 | \
Gamma(3/4)* | | | -1|
2 1 \ 7/4 | /
--------------------------------
4*Gamma(7/4)
$$\frac{\Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {-1} \right)}}{4 \Gamma\left(\frac{7}{4}\right)}$$
=
=
_
|_ /-1/2, 3/4 | \
Gamma(3/4)* | | | -1|
2 1 \ 7/4 | /
--------------------------------
4*Gamma(7/4)
$$\frac{\Gamma\left(\frac{3}{4}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle| {-1} \right)}}{4 \Gamma\left(\frac{7}{4}\right)}$$
gamma(3/4)*hyper((-1/2, 3/4), (7/4,), -1)/(4*gamma(7/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.