Sr Examen
Integral de x^2/(x^6-1)^1/2 dx
Solución
Ha introducido
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1 / | | 2 | x | ----------- dx | ________ | / 6 | \/ x - 1 | / 0
$$\int\limits_{0}^{1} \frac{x^{2}}{\sqrt{x^{6} - 1}}\, dx$$
Integral(x^2/sqrt(x^6 - 1), (x, 0, 1))
Respuesta (Indefinida)
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/ // / 3\ \ | || acosh\x / | 6| | | 2 || --------- for |x | > 1| | x || 3 | | ----------- dx = C + |< | | ________ || / 3\ | | / 6 ||-I*asin\x / | | \/ x - 1 ||------------ otherwise | | \\ 3 / /
$$\int \frac{x^{2}}{\sqrt{x^{6} - 1}}\, dx = C + \begin{cases} \frac{\operatorname{acosh}{\left(x^{3} \right)}}{3} & \text{for}\: \left|{x^{6}}\right| > 1 \\- \frac{i \operatorname{asin}{\left(x^{3} \right)}}{3} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 2 | | x 6 | |------------ for x > 1 | | _________ | | / 6 | |\/ -1 + x | < dx | | 2 | | -I*x | |----------- otherwise | | ________ | | / 6 | \\/ 1 - x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{x^{2}}{\sqrt{x^{6} - 1}} & \text{for}\: x^{6} > 1 \\- \frac{i x^{2}}{\sqrt{1 - x^{6}}} & \text{otherwise} \end{cases}\, dx$$
=
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1 / | | / 2 | | x 6 | |------------ for x > 1 | | _________ | | / 6 | |\/ -1 + x | < dx | | 2 | | -I*x | |----------- otherwise | | ________ | | / 6 | \\/ 1 - x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{x^{2}}{\sqrt{x^{6} - 1}} & \text{for}\: x^{6} > 1 \\- \frac{i x^{2}}{\sqrt{1 - x^{6}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((x^2/sqrt(-1 + x^6), x^6 > 1), (-i*x^2/sqrt(1 - x^6), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.