Sr Examen
Integral de x^3*dx/(a^2+x^6) dx
Solución
Ha introducido
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oo / | | 3 | x | ------- dx | 2 6 | a + x | / 0
$$\int\limits_{0}^{\infty} \frac{x^{3}}{a^{2} + x^{6}}\, dx$$
Integral(x^3/(a^2 + x^6), (x, 0, oo))
Respuesta (Indefinida)
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/ | | 3 | x / 2 3 / 2 2 2\\ | ------- dx = C + RootSum\216*a *t + 1, t -> t*log\x + 36*a *t // | 2 6 | a + x | /
$$\int \frac{x^{3}}{a^{2} + x^{6}}\, dx = C + \operatorname{RootSum} {\left(216 t^{3} a^{2} + 1, \left( t \mapsto t \log{\left(36 t^{2} a^{2} + x^{2} \right)} \right)\right)}$$
Respuesta
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/ ___ 2 | pi*\/ 3 *a / / 1 \ \ |------------------ for Or|And|2*|arg(a)| < pi, -- != 0, 2*|arg(a)| != pi|, 2*|arg(a)| < pi| | 8/3 | | 2 | | |9*polar_lift (a) \ \ a / / | | oo | / < | | | 3 | | x | | ------- dx otherwise | | 2 6 | | a + x | | | / \ 0
$$\begin{cases} \frac{\sqrt{3} \pi a^{2}}{9 \operatorname{polar\_lift}^{\frac{8}{3}}{\left(a \right)}} & \text{for}\: \left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge \frac{1}{a^{2}} \neq 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{x^{3}}{a^{2} + x^{6}}\, dx & \text{otherwise} \end{cases}$$
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/ ___ 2 | pi*\/ 3 *a / / 1 \ \ |------------------ for Or|And|2*|arg(a)| < pi, -- != 0, 2*|arg(a)| != pi|, 2*|arg(a)| < pi| | 8/3 | | 2 | | |9*polar_lift (a) \ \ a / / | | oo | / < | | | 3 | | x | | ------- dx otherwise | | 2 6 | | a + x | | | / \ 0
$$\begin{cases} \frac{\sqrt{3} \pi a^{2}}{9 \operatorname{polar\_lift}^{\frac{8}{3}}{\left(a \right)}} & \text{for}\: \left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge \frac{1}{a^{2}} \neq 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{0}^{\infty} \frac{x^{3}}{a^{2} + x^{6}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((pi*sqrt(3)*a^2/(9*polar_lift(a)^(8/3)), (2*Abs(arg(a)) < pi)∨((Ne(a^(-2), 0))∧(2*Abs(arg(a)) < pi)∧(Ne(2*Abs(arg(a)), pi)))), (Integral(x^3/(a^2 + x^6), (x, 0, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.