Sr Examen
Integral de (x^3-1)^(1/2) dx
Solución
Ha introducido
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2 / | | ________ | / 3 | \/ x - 1 dx | / 1
$$\int\limits_{1}^{2} \sqrt{x^{3} - 1}\, dx$$
Integral(sqrt(x^3 - 1), (x, 1, 2))
Respuesta (Indefinida)
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/ _ | |_ /-1/2, 1/3 | 3\ | ________ I*x*Gamma(1/3)* | | | x | | / 3 2 1 \ 4/3 | / | \/ x - 1 dx = C + ------------------------------------ | 3*Gamma(4/3) /
$$\int \sqrt{x^{3} - 1}\, dx = C + \frac{i x \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {x^{3}} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$
Gráfica
Respuesta
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_ _
|_ /-1/2, 1/3 | \ |_ /-1/2, 1/3 | \
I*Gamma(1/3)* | | | 1| 2*I*Gamma(1/3)* | | | 8|
2 1 \ 4/3 | / 2 1 \ 4/3 | /
- --------------------------------- + -----------------------------------
3*Gamma(4/3) 3*Gamma(4/3)
$$- \frac{i \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {1} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{2 i \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {8} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$
=
=
_ _
|_ /-1/2, 1/3 | \ |_ /-1/2, 1/3 | \
I*Gamma(1/3)* | | | 1| 2*I*Gamma(1/3)* | | | 8|
2 1 \ 4/3 | / 2 1 \ 4/3 | /
- --------------------------------- + -----------------------------------
3*Gamma(4/3) 3*Gamma(4/3)
$$- \frac{i \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {1} \right)}}{3 \Gamma\left(\frac{4}{3}\right)} + \frac{2 i \Gamma\left(\frac{1}{3}\right) {{}_{2}F_{1}\left(\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle| {8} \right)}}{3 \Gamma\left(\frac{4}{3}\right)}$$
-i*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 1)/(3*gamma(4/3)) + 2*i*gamma(1/3)*hyper((-1/2, 1/3), (4/3,), 8)/(3*gamma(4/3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.