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