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