Sr Examen
Integral de (x-((x^2)*sqrt(x+3)))/(x^(1/3)) dx
Solución
Ha introducido
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1 / | | 2 _______ | x - x *\/ x + 3 | ---------------- dx | 3 ___ | \/ x | / 0
$$\int\limits_{0}^{1} \frac{- x^{2} \sqrt{x + 3} + x}{\sqrt[3]{x}}\, dx$$
Integral((x - x^2*sqrt(x + 3))/x^(1/3), (x, 0, 1))
Respuesta (Indefinida)
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2*pi*I / ------ _ | 2*pi*I 2/3 5/2 3 |_ /-2/3, 5/2 | x\ | 2 _______ 5/3 ------ _ 2*3 *(3 + x) *e * | | | 1 + -| | x - x *\/ x + 3 3*x 2/3 3/2 3 |_ /-2/3, 3/2 | x\ 2 1 \ 7/2 | 3/ | ---------------- dx = C + ------ + 2*3 *(3 + x) *e * | | | 1 + -| - -------------------------------------------------- | 3 ___ 5 2 1 \ 5/2 | 3/ 5 | \/ x | /
$$\int \frac{- x^{2} \sqrt{x + 3} + x}{\sqrt[3]{x}}\, dx = C + \frac{3 x^{\frac{5}{3}}}{5} - \frac{2 \cdot 3^{\frac{2}{3}} \left(x + 3\right)^{\frac{5}{2}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle| {\frac{x}{3} + 1} \right)}}{5} + 2 \cdot 3^{\frac{2}{3}} \left(x + 3\right)^{\frac{3}{2}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle| {\frac{x}{3} + 1} \right)}$$
Gráfica
Respuesta
[src]
2*pi*I 2*pi*I
------ _ ------ _
2*pi*I 2*pi*I 2/3 3 |_ /-2/3, 5/2 | \ 6 ___ 3 |_ /-2/3, 5/2 | \
------ _ ------ _ 64*3 *e * | | | 4/3| 54*\/ 3 *e * | | | 1|
3 6 ___ 3 |_ /-2/3, 3/2 | \ 2/3 3 |_ /-2/3, 3/2 | \ 2 1 \ 7/2 | / 2 1 \ 7/2 | /
- - 18*\/ 3 *e * | | | 1| + 16*3 *e * | | | 4/3| - -------------------------------------- + -------------------------------------
5 2 1 \ 5/2 | / 2 1 \ 5/2 | / 5 5
$$- 18 \sqrt[6]{3} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle| {1} \right)} + 16 \cdot 3^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle| {\frac{4}{3}} \right)} + \frac{54 \sqrt[6]{3} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle| {1} \right)}}{5} - \frac{64 \cdot 3^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle| {\frac{4}{3}} \right)}}{5} + \frac{3}{5}$$
=
=
2*pi*I 2*pi*I
------ _ ------ _
2*pi*I 2*pi*I 2/3 3 |_ /-2/3, 5/2 | \ 6 ___ 3 |_ /-2/3, 5/2 | \
------ _ ------ _ 64*3 *e * | | | 4/3| 54*\/ 3 *e * | | | 1|
3 6 ___ 3 |_ /-2/3, 3/2 | \ 2/3 3 |_ /-2/3, 3/2 | \ 2 1 \ 7/2 | / 2 1 \ 7/2 | /
- - 18*\/ 3 *e * | | | 1| + 16*3 *e * | | | 4/3| - -------------------------------------- + -------------------------------------
5 2 1 \ 5/2 | / 2 1 \ 5/2 | / 5 5
$$- 18 \sqrt[6]{3} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle| {1} \right)} + 16 \cdot 3^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle| {\frac{4}{3}} \right)} + \frac{54 \sqrt[6]{3} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle| {1} \right)}}{5} - \frac{64 \cdot 3^{\frac{2}{3}} e^{\frac{2 i \pi}{3}} {{}_{2}F_{1}\left(\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle| {\frac{4}{3}} \right)}}{5} + \frac{3}{5}$$
3/5 - 18*3^(1/6)*exp(2*pi*i/3)*hyper((-2/3, 3/2), (5/2,), 1) + 16*3^(2/3)*exp(2*pi*i/3)*hyper((-2/3, 3/2), (5/2,), 4/3) - 64*3^(2/3)*exp(2*pi*i/3)*hyper((-2/3, 5/2), (7/2,), 4/3)/5 + 54*3^(1/6)*exp(2*pi*i/3)*hyper((-2/3, 5/2), (7/2,), 1)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.