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