Sr Examen
Integral de 1/(sqrt(abs(x^2-1))) dx
Solución
Ha introducido
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1 / | | 1 | ------------- dx | __________ | / | 2 | | \/ |x - 1| | / 0
$$\int\limits_{0}^{1} \frac{1}{\sqrt{\left|{x^{2} - 1}\right|}}\, dx$$
Integral(1/(sqrt(|x^2 - 1|)), (x, 0, 1))
Respuesta (Indefinida)
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// ___ \
|| pi*\/ 2 *asin(x) | 2| |
/ || --------------------- for |x | < 1|
| || Gamma(1/4)*Gamma(3/4) |
| 1 || |
| ------------- dx = C + |< ___ ____ __1, 2 /1, 1 3/4 | 2\ |
| __________ ||\/ 2 *\/ pi */__ | | x | |
| / | 2 | || \_|3, 3 \1/2 3/4, 0 | / |
| \/ |x - 1| ||---------------------------------------- otherwise |
| || 2 |
/ \\ /
$$\int \frac{1}{\sqrt{\left|{x^{2} - 1}\right|}}\, dx = C + \begin{cases} \frac{\sqrt{2} \pi \operatorname{asin}{\left(x \right)}}{\Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} & \text{for}\: \left|{x^{2}}\right| < 1 \\\frac{\sqrt{2} \sqrt{\pi} {G_{3, 3}^{1, 2}\left(\begin{matrix} 1, 1 & \frac{3}{4} \\\frac{1}{2} & \frac{3}{4}, 0 \end{matrix} \middle| {x^{2}} \right)}}{2} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 1 2 | |------------ for -1 + x >= 0 | | _________ | | / 2 | |\/ -1 + x | < dx | | 1 | |----------- otherwise | | ________ | | / 2 | \\/ 1 - x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{\sqrt{x^{2} - 1}} & \text{for}\: x^{2} - 1 \geq 0 \\\frac{1}{\sqrt{1 - x^{2}}} & \text{otherwise} \end{cases}\, dx$$
=
=
1 / | | / 1 2 | |------------ for -1 + x >= 0 | | _________ | | / 2 | |\/ -1 + x | < dx | | 1 | |----------- otherwise | | ________ | | / 2 | \\/ 1 - x | / 0
$$\int\limits_{0}^{1} \begin{cases} \frac{1}{\sqrt{x^{2} - 1}} & \text{for}\: x^{2} - 1 \geq 0 \\\frac{1}{\sqrt{1 - x^{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((1/sqrt(-1 + x^2), -1 + x^2 >= 0), (1/sqrt(1 - x^2), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.