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