Sr Examen
Integral de x^2*(16-x^2)^(1/2) dx
Solución
Ha introducido
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pi -- 2 / | | _________ | 2 / 2 | x *\/ 16 - x dx | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} x^{2} \sqrt{16 - x^{2}}\, dx$$
Integral(x^2*sqrt(16 - x^2), (x, 0, pi/2))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
TrigSubstitutionRule(theta=_theta, func=4*sin(_theta), rewritten=32 - 32*cos(4*_theta), substep=AddRule(substeps=[ConstantRule(constant=32, context=32, symbol=_theta), ConstantTimesRule(constant=-32, 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=-32*cos(4*_theta), symbol=_theta)], context=32 - 32*cos(4*_theta), symbol=_theta), restriction=(x > -4) & (x < 4), context=x**2*sqrt(16 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | _________ // _________ \ | 2 / 2 || / 2 / 2\ | | x *\/ 16 - x dx = C + |< /x\ x*\/ 16 - x *\8 - x / | | ||32*asin|-| - ----------------------- for And(x > -4, x < 4)| / \\ \4/ 4 /
$$\int x^{2} \sqrt{16 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(8 - x^{2}\right) \sqrt{16 - x^{2}}}{4} + 32 \operatorname{asin}{\left(\frac{x}{4} \right)} & \text{for}\: x > -4 \wedge x < 4 \end{cases}$$
Gráfica
Respuesta
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pi -- 2 / | | / 2 2 4 6 4 2 | | 8*I 32*I 32*I*x 18*I*x 6*I*x I*x 5*I*x x | |- -------------- + ------------- - ------------- - ------------- + ------------- - --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 3/2 __________ 16 | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | / x \/ -16 + x \-16 + x / \/ -16 + x \-16 + x / 4*\-16 + x / 4*\/ -16 + x | | / -1 + -- | | \/ 16 | < dx | | 2 4 2 4 6 | | 32 8 32*x 6*x 18*x 5*x x | | - ------------ + ------------- - ------------ + ------------ + ------------ - -------------- - -------------- otherwise | | _________ ________ 3/2 3/2 _________ _________ 3/2 | | / 2 / 2 / 2\ / 2\ / 2 / 2 / 2\ | | \/ 16 - x / x \16 - x / \16 - x / \/ 16 - x 4*\/ 16 - x 4*\16 - x / | | / 1 - -- | \ \/ 16 | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 16\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 16}} + \frac{6 i x^{4}}{\left(x^{2} - 16\right)^{\frac{3}{2}}} - \frac{18 i x^{2}}{\sqrt{x^{2} - 16}} - \frac{32 i x^{2}}{\left(x^{2} - 16\right)^{\frac{3}{2}}} + \frac{32 i}{\sqrt{x^{2} - 16}} - \frac{8 i}{\sqrt{\frac{x^{2}}{16} - 1}} & \text{for}\: \frac{x^{2}}{16} > 1 \\- \frac{x^{6}}{4 \left(16 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{16 - x^{2}}} + \frac{6 x^{4}}{\left(16 - x^{2}\right)^{\frac{3}{2}}} + \frac{18 x^{2}}{\sqrt{16 - x^{2}}} - \frac{32 x^{2}}{\left(16 - x^{2}\right)^{\frac{3}{2}}} - \frac{32}{\sqrt{16 - x^{2}}} + \frac{8}{\sqrt{1 - \frac{x^{2}}{16}}} & \text{otherwise} \end{cases}\, dx$$
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pi -- 2 / | | / 2 2 4 6 4 2 | | 8*I 32*I 32*I*x 18*I*x 6*I*x I*x 5*I*x x | |- -------------- + ------------- - ------------- - ------------- + ------------- - --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 3/2 __________ 16 | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | / x \/ -16 + x \-16 + x / \/ -16 + x \-16 + x / 4*\-16 + x / 4*\/ -16 + x | | / -1 + -- | | \/ 16 | < dx | | 2 4 2 4 6 | | 32 8 32*x 6*x 18*x 5*x x | | - ------------ + ------------- - ------------ + ------------ + ------------ - -------------- - -------------- otherwise | | _________ ________ 3/2 3/2 _________ _________ 3/2 | | / 2 / 2 / 2\ / 2\ / 2 / 2 / 2\ | | \/ 16 - x / x \16 - x / \16 - x / \/ 16 - x 4*\/ 16 - x 4*\16 - x / | | / 1 - -- | \ \/ 16 | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 16\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 16}} + \frac{6 i x^{4}}{\left(x^{2} - 16\right)^{\frac{3}{2}}} - \frac{18 i x^{2}}{\sqrt{x^{2} - 16}} - \frac{32 i x^{2}}{\left(x^{2} - 16\right)^{\frac{3}{2}}} + \frac{32 i}{\sqrt{x^{2} - 16}} - \frac{8 i}{\sqrt{\frac{x^{2}}{16} - 1}} & \text{for}\: \frac{x^{2}}{16} > 1 \\- \frac{x^{6}}{4 \left(16 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{16 - x^{2}}} + \frac{6 x^{4}}{\left(16 - x^{2}\right)^{\frac{3}{2}}} + \frac{18 x^{2}}{\sqrt{16 - x^{2}}} - \frac{32 x^{2}}{\left(16 - x^{2}\right)^{\frac{3}{2}}} - \frac{32}{\sqrt{16 - x^{2}}} + \frac{8}{\sqrt{1 - \frac{x^{2}}{16}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-8*i/sqrt(-1 + x^2/16) + 32*i/sqrt(-16 + x^2) - 32*i*x^2/(-16 + x^2)^(3/2) - 18*i*x^2/sqrt(-16 + x^2) + 6*i*x^4/(-16 + x^2)^(3/2) - i*x^6/(4*(-16 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-16 + x^2)), x^2/16 > 1), (-32/sqrt(16 - x^2) + 8/sqrt(1 - x^2/16) - 32*x^2/(16 - x^2)^(3/2) + 6*x^4/(16 - x^2)^(3/2) + 18*x^2/sqrt(16 - x^2) - 5*x^4/(4*sqrt(16 - x^2)) - x^6/(4*(16 - x^2)^(3/2)), True)), (x, 0, pi/2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.