Sr Examen
Integral de 1/(sqrt(c-x^(2/3))) dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | __________ | / 2/3 | \/ c - x | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{c - x^{\frac{2}{3}}}}\, dx$$
Integral(1/(sqrt(c - x^(2/3))), (x, 0, 1))
Respuesta (Indefinida)
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// /3 ___\ ___________ \
|| |\/ x | / 2/3 |
|| 3*I*c*acosh|-----| ___ 3 ___ / x |
|| | ___| 3*I*\/ c *\/ x * / -1 + ---- | 2/3| |
|| \\/ c / \/ c |x | |
|| - ------------------ - -------------------------------- for |----| > 1|
/ || 2 2 | c | |
| || |
| 1 || /3 ___\ |
| ------------- dx = C + |< |\/ x | |
| __________ ||3*c*asin|-----| |
| / 2/3 || | ___| ___ 3 ___ |
| \/ c - x || \\/ c / 3*\/ c *\/ x 3*x |
| ||--------------- - ----------------- + ----------------------- otherwise |
/ || 2 __________ __________ |
|| / 2/3 / 2/3 |
|| / x ___ / x |
|| 2* / 1 - ---- 2*\/ c * / 1 - ---- |
\\ \/ c \/ c /
$$\int \frac{1}{\sqrt{c - x^{\frac{2}{3}}}}\, dx = C + \begin{cases} - \frac{3 i \sqrt{c} \sqrt[3]{x} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}}{2} - \frac{3 i c \operatorname{acosh}{\left(\frac{\sqrt[3]{x}}{\sqrt{c}} \right)}}{2} & \text{for}\: \left|{\frac{x^{\frac{2}{3}}}{c}}\right| > 1 \\- \frac{3 \sqrt{c} \sqrt[3]{x}}{2 \sqrt{1 - \frac{x^{\frac{2}{3}}}{c}}} + \frac{3 c \operatorname{asin}{\left(\frac{\sqrt[3]{x}}{\sqrt{c}} \right)}}{2} + \frac{3 x}{2 \sqrt{c} \sqrt{1 - \frac{x^{\frac{2}{3}}}{c}}} & \text{otherwise} \end{cases}$$
Respuesta
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1 / | | / ___________ | | / 2/3 | | ___ / x | | I*\/ c * / -1 + ---- ___ 2/3 | | I \/ c I*\/ c x | |- ------------------------ - ------------------------ - ----------------------- for ---- > 1 | | ___________ 2/3 ___________ |c| | | / 2/3 2*x / 2/3 | | ___ / x 2/3 / x | | 2*\/ c * / -1 + ---- 2*x * / -1 + ---- | < \/ c \/ c dx | | | | 2/3 | | 1 3 x | | - --------------------- + ----------------------- + -------------------- otherwise | | 3/2 __________ 3/2 | | / 2/3\ / 2/3 / 2/3\ | | ___ | x | ___ / x 3/2 | x | | | 2*\/ c *|1 - ----| 2*\/ c * / 1 - ---- 2*c *|1 - ----| | | \ c / \/ c \ c / | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{c} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}}{2 x^{\frac{2}{3}}} - \frac{i \sqrt{c}}{2 x^{\frac{2}{3}} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}} - \frac{i}{2 \sqrt{c} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}} & \text{for}\: \frac{x^{\frac{2}{3}}}{\left|{c}\right|} > 1 \\\frac{3}{2 \sqrt{c} \sqrt{1 - \frac{x^{\frac{2}{3}}}{c}}} - \frac{1}{2 \sqrt{c} \left(1 - \frac{x^{\frac{2}{3}}}{c}\right)^{\frac{3}{2}}} + \frac{x^{\frac{2}{3}}}{2 c^{\frac{3}{2}} \left(1 - \frac{x^{\frac{2}{3}}}{c}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / ___________ | | / 2/3 | | ___ / x | | I*\/ c * / -1 + ---- ___ 2/3 | | I \/ c I*\/ c x | |- ------------------------ - ------------------------ - ----------------------- for ---- > 1 | | ___________ 2/3 ___________ |c| | | / 2/3 2*x / 2/3 | | ___ / x 2/3 / x | | 2*\/ c * / -1 + ---- 2*x * / -1 + ---- | < \/ c \/ c dx | | | | 2/3 | | 1 3 x | | - --------------------- + ----------------------- + -------------------- otherwise | | 3/2 __________ 3/2 | | / 2/3\ / 2/3 / 2/3\ | | ___ | x | ___ / x 3/2 | x | | | 2*\/ c *|1 - ----| 2*\/ c * / 1 - ---- 2*c *|1 - ----| | | \ c / \/ c \ c / | \ | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i \sqrt{c} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}}{2 x^{\frac{2}{3}}} - \frac{i \sqrt{c}}{2 x^{\frac{2}{3}} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}} - \frac{i}{2 \sqrt{c} \sqrt{-1 + \frac{x^{\frac{2}{3}}}{c}}} & \text{for}\: \frac{x^{\frac{2}{3}}}{\left|{c}\right|} > 1 \\\frac{3}{2 \sqrt{c} \sqrt{1 - \frac{x^{\frac{2}{3}}}{c}}} - \frac{1}{2 \sqrt{c} \left(1 - \frac{x^{\frac{2}{3}}}{c}\right)^{\frac{3}{2}}} + \frac{x^{\frac{2}{3}}}{2 c^{\frac{3}{2}} \left(1 - \frac{x^{\frac{2}{3}}}{c}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i/(2*sqrt(c)*sqrt(-1 + x^(2/3)/c)) - i*sqrt(c)*sqrt(-1 + x^(2/3)/c)/(2*x^(2/3)) - i*sqrt(c)/(2*x^(2/3)*sqrt(-1 + x^(2/3)/c)), x^(2/3)/|c| > 1), (-1/(2*sqrt(c)*(1 - x^(2/3)/c)^(3/2)) + 3/(2*sqrt(c)*sqrt(1 - x^(2/3)/c)) + x^(2/3)/(2*c^(3/2)*(1 - x^(2/3)/c)^(3/2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.