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