Sr Examen
Integral de x^2*(8-x^2)^(1/2) dx
Solución
Ha introducido
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___ \/ 2 / | | ________ | 2 / 2 | x *\/ 8 - x dx | / 0
$$\int\limits_{0}^{\sqrt{2}} x^{2} \sqrt{8 - x^{2}}\, dx$$
Integral(x^2*sqrt(8 - x^2), (x, 0, sqrt(2)))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
TrigSubstitutionRule(theta=_theta, func=2*sqrt(2)*sin(_theta), rewritten=8 - 8*cos(4*_theta), substep=AddRule(substeps=[ConstantRule(constant=8, context=8, symbol=_theta), ConstantTimesRule(constant=-8, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=-8*cos(4*_theta), symbol=_theta)], context=8 - 8*cos(4*_theta), symbol=_theta), restriction=(x > -2*sqrt(2)) & (x < 2*sqrt(2)), context=x**2*sqrt(8 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | ________ // ________ \ | 2 / 2 || / ___\ / 2 / 2\ | | x *\/ 8 - x dx = C + |< |x*\/ 2 | x*\/ 8 - x *\4 - x / / ___ ___\| | ||8*asin|-------| - ---------------------- for And\x > -2*\/ 2 , x < 2*\/ 2 /| / \\ \ 4 / 4 /
$$\int x^{2} \sqrt{8 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(4 - x^{2}\right) \sqrt{8 - x^{2}}}{4} + 8 \operatorname{asin}{\left(\frac{\sqrt{2} x}{4} \right)} & \text{for}\: x > - 2 \sqrt{2} \wedge x < 2 \sqrt{2} \end{cases}$$
Gráfica
Respuesta
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___ \/ 2 / | | / 2 2 ___ 4 6 4 2 | | 8*I 9*I*x 8*I*x 2*I*\/ 2 3*I*x I*x 5*I*x x | |------------ - ------------ - ------------ - -------------- + ------------ - -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 3/2 _________ 8 | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | |\/ -8 + x \/ -8 + x \-8 + x / / x \-8 + x / 4*\-8 + x / 4*\/ -8 + x | | / -1 + -- | | \/ 8 | < dx | | 2 ___ 4 2 4 6 | | 8 8*x 2*\/ 2 3*x 9*x 5*x x | | - ----------- - ----------- + ------------- + ----------- + ----------- - ------------- - ------------- otherwise | | ________ 3/2 ________ 3/2 ________ ________ 3/2 | | / 2 / 2\ / 2 / 2\ / 2 / 2 / 2\ | | \/ 8 - x \8 - x / / x \8 - x / \/ 8 - x 4*\/ 8 - x 4*\8 - x / | | / 1 - -- | \ \/ 8 | / 0
$$\int\limits_{0}^{\sqrt{2}} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 8\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 8}} + \frac{3 i x^{4}}{\left(x^{2} - 8\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{\sqrt{x^{2} - 8}} - \frac{8 i x^{2}}{\left(x^{2} - 8\right)^{\frac{3}{2}}} + \frac{8 i}{\sqrt{x^{2} - 8}} - \frac{2 \sqrt{2} i}{\sqrt{\frac{x^{2}}{8} - 1}} & \text{for}\: \frac{x^{2}}{8} > 1 \\- \frac{x^{6}}{4 \left(8 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{8 - x^{2}}} + \frac{3 x^{4}}{\left(8 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{\sqrt{8 - x^{2}}} - \frac{8 x^{2}}{\left(8 - x^{2}\right)^{\frac{3}{2}}} - \frac{8}{\sqrt{8 - x^{2}}} + \frac{2 \sqrt{2}}{\sqrt{1 - \frac{x^{2}}{8}}} & \text{otherwise} \end{cases}\, dx$$
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___ \/ 2 / | | / 2 2 ___ 4 6 4 2 | | 8*I 9*I*x 8*I*x 2*I*\/ 2 3*I*x I*x 5*I*x x | |------------ - ------------ - ------------ - -------------- + ------------ - -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 3/2 _________ 8 | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | |\/ -8 + x \/ -8 + x \-8 + x / / x \-8 + x / 4*\-8 + x / 4*\/ -8 + x | | / -1 + -- | | \/ 8 | < dx | | 2 ___ 4 2 4 6 | | 8 8*x 2*\/ 2 3*x 9*x 5*x x | | - ----------- - ----------- + ------------- + ----------- + ----------- - ------------- - ------------- otherwise | | ________ 3/2 ________ 3/2 ________ ________ 3/2 | | / 2 / 2\ / 2 / 2\ / 2 / 2 / 2\ | | \/ 8 - x \8 - x / / x \8 - x / \/ 8 - x 4*\/ 8 - x 4*\8 - x / | | / 1 - -- | \ \/ 8 | / 0
$$\int\limits_{0}^{\sqrt{2}} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 8\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 8}} + \frac{3 i x^{4}}{\left(x^{2} - 8\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{\sqrt{x^{2} - 8}} - \frac{8 i x^{2}}{\left(x^{2} - 8\right)^{\frac{3}{2}}} + \frac{8 i}{\sqrt{x^{2} - 8}} - \frac{2 \sqrt{2} i}{\sqrt{\frac{x^{2}}{8} - 1}} & \text{for}\: \frac{x^{2}}{8} > 1 \\- \frac{x^{6}}{4 \left(8 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{8 - x^{2}}} + \frac{3 x^{4}}{\left(8 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{\sqrt{8 - x^{2}}} - \frac{8 x^{2}}{\left(8 - x^{2}\right)^{\frac{3}{2}}} - \frac{8}{\sqrt{8 - x^{2}}} + \frac{2 \sqrt{2}}{\sqrt{1 - \frac{x^{2}}{8}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((8*i/sqrt(-8 + x^2) - 9*i*x^2/sqrt(-8 + x^2) - 8*i*x^2/(-8 + x^2)^(3/2) - 2*i*sqrt(2)/sqrt(-1 + x^2/8) + 3*i*x^4/(-8 + x^2)^(3/2) - i*x^6/(4*(-8 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-8 + x^2)), x^2/8 > 1), (-8/sqrt(8 - x^2) - 8*x^2/(8 - x^2)^(3/2) + 2*sqrt(2)/sqrt(1 - x^2/8) + 3*x^4/(8 - x^2)^(3/2) + 9*x^2/sqrt(8 - x^2) - 5*x^4/(4*sqrt(8 - x^2)) - x^6/(4*(8 - x^2)^(3/2)), True)), (x, 0, sqrt(2)))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.