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