Sr Examen
Integral de 1/(y^(3/2)+c) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------- dy | 3/2 | y + c | / 0
$$\int\limits_{0}^{1} \frac{1}{c + y^{\frac{3}{2}}}\, dy$$
Integral(1/(y^(3/2) + c), (y, 0, 1))
Respuesta (Indefinida)
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// / ___ 2/3 ___ ___\ \
|| 2/3 ___ |\/ 3 2*(-1) *\/ 3 *\/ y | |
|| 2*(-1) *\/ 3 *atan|----- - ---------------------| |
/ || 2/3 / ___ 3 ____ 3 ___\ 2/3 / 2/3 2/3 3 ____ 3 ___ ___\ | 3 3 ___ | |
| || 2*(-1) *log\\/ y - \/ -1 *\/ c / (-1) *log\4*y + 4*(-1) *c + 4*\/ -1 *\/ c *\/ y / \ 3*\/ c / |
| 1 ||- ----------------------------------- + -------------------------------------------------------- - --------------------------------------------------- for c != 0|
| -------- dy = C + |< 3 ___ 3 ___ 3 ___ |
| 3/2 || 3*\/ c 3*\/ c 3*\/ c |
| y + c || |
| || -2 |
/ || ----- otherwise |
|| ___ |
\\ \/ y /
$$\int \frac{1}{c + y^{\frac{3}{2}}}\, dy = C + \begin{cases} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + \sqrt{y} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} \sqrt{y} + 4 y \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \sqrt{y}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} & \text{for}\: c \neq 0 \\- \frac{2}{\sqrt{y}} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ / ___ 2/3 ___\ | 2/3 ___ |\/ 3 2*(-1) *\/ 3 | | 2*(-1) *\/ 3 *atan|----- - ---------------| | 2/3 / 3 ____ 3 ___\ 2/3 / 2/3 2/3\ 2/3 / 3 ____ 3 ___ 2/3 2/3\ 2/3 / 3 ____ 3 ___\ | 3 3 ___ | 2/3 ___ | 2*(-1) *log\1 - \/ -1 *\/ c / (-1) *log\4*(-1) *c / (-1) *log\4 + 4*\/ -1 *\/ c + 4*(-1) *c / 2*(-1) *log\-\/ -1 *\/ c / \ 3*\/ c / pi*(-1) *\/ 3 <- ------------------------------- - --------------------------- + ------------------------------------------------ + ---------------------------- - --------------------------------------------- + ---------------- for And(c > -oo, c < oo, c != 0) | 3 ___ 3 ___ 3 ___ 3 ___ 3 ___ 3 ___ | 3*\/ c 3*\/ c 3*\/ c 3*\/ c 3*\/ c 9*\/ c | | oo otherwise \
$$\begin{cases} \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} \right)}}{3 \sqrt[3]{c}} - \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + 1 \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} + 4 \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \sqrt{3} \pi}{9 \sqrt[3]{c}} & \text{for}\: c > -\infty \wedge c < \infty \wedge c \neq 0 \\\infty & \text{otherwise} \end{cases}$$
=
=
/ / ___ 2/3 ___\ | 2/3 ___ |\/ 3 2*(-1) *\/ 3 | | 2*(-1) *\/ 3 *atan|----- - ---------------| | 2/3 / 3 ____ 3 ___\ 2/3 / 2/3 2/3\ 2/3 / 3 ____ 3 ___ 2/3 2/3\ 2/3 / 3 ____ 3 ___\ | 3 3 ___ | 2/3 ___ | 2*(-1) *log\1 - \/ -1 *\/ c / (-1) *log\4*(-1) *c / (-1) *log\4 + 4*\/ -1 *\/ c + 4*(-1) *c / 2*(-1) *log\-\/ -1 *\/ c / \ 3*\/ c / pi*(-1) *\/ 3 <- ------------------------------- - --------------------------- + ------------------------------------------------ + ---------------------------- - --------------------------------------------- + ---------------- for And(c > -oo, c < oo, c != 0) | 3 ___ 3 ___ 3 ___ 3 ___ 3 ___ 3 ___ | 3*\/ c 3*\/ c 3*\/ c 3*\/ c 3*\/ c 9*\/ c | | oo otherwise \
$$\begin{cases} \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} \right)}}{3 \sqrt[3]{c}} - \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \log{\left(- \sqrt[3]{-1} \sqrt[3]{c} + 1 \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \log{\left(4 \left(-1\right)^{\frac{2}{3}} c^{\frac{2}{3}} + 4 \sqrt[3]{-1} \sqrt[3]{c} + 4 \right)}}{3 \sqrt[3]{c}} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{3} - \frac{2 \left(-1\right)^{\frac{2}{3}} \sqrt{3}}{3 \sqrt[3]{c}} \right)}}{3 \sqrt[3]{c}} + \frac{\left(-1\right)^{\frac{2}{3}} \sqrt{3} \pi}{9 \sqrt[3]{c}} & \text{for}\: c > -\infty \wedge c < \infty \wedge c \neq 0 \\\infty & \text{otherwise} \end{cases}$$
Piecewise((-2*(-1)^(2/3)*log(1 - (-1)^(1/3)*c^(1/3))/(3*c^(1/3)) - (-1)^(2/3)*log(4*(-1)^(2/3)*c^(2/3))/(3*c^(1/3)) + (-1)^(2/3)*log(4 + 4*(-1)^(1/3)*c^(1/3) + 4*(-1)^(2/3)*c^(2/3))/(3*c^(1/3)) + 2*(-1)^(2/3)*log(-(-1)^(1/3)*c^(1/3))/(3*c^(1/3)) - 2*(-1)^(2/3)*sqrt(3)*atan(sqrt(3)/3 - 2*(-1)^(2/3)*sqrt(3)/(3*c^(1/3)))/(3*c^(1/3)) + pi*(-1)^(2/3)*sqrt(3)/(9*c^(1/3)), (c > -oo)∧(c < oo)∧(Ne(c, 0))), (oo, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.