Sr Examen
Integral de x^2sqrt(9-x^2) dx
Solución
Ha introducido
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3 / | | ________ | 2 / 2 | x *\/ 9 - x dx | / 0
$$\int\limits_{0}^{3} x^{2} \sqrt{9 - x^{2}}\, dx$$
Integral(x^2*sqrt(9 - x^2), (x, 0, 3))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
TrigSubstitutionRule(theta=_theta, func=3*sin(_theta), rewritten=81/8 - 81*cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=81/8, context=81/8, symbol=_theta), ConstantTimesRule(constant=-81/8, 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=-81*cos(4*_theta)/8, symbol=_theta)], context=81/8 - 81*cos(4*_theta)/8, symbol=_theta), restriction=(x > -3) & (x < 3), context=x**2*sqrt(9 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | ________ // /x\ ________ \ | 2 / 2 ||81*asin|-| / 2 / 2\ | | x *\/ 9 - x dx = C + |< \3/ x*\/ 9 - x *\9 - 2*x / | | ||---------- - ------------------------ for And(x > -3, x < 3)| / \\ 8 8 /
$$\int x^{2} \sqrt{9 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(9 - 2 x^{2}\right) \sqrt{9 - x^{2}}}{8} + \frac{81 \operatorname{asin}{\left(\frac{x}{3} \right)}}{8} & \text{for}\: x > -3 \wedge x < 3 \end{cases}$$
Gráfica
Respuesta
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3 / | | / 2 2 6 4 4 2 | | 27*I 81*I 81*I*x 81*I*x I*x 5*I*x 27*I*x x | |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 9 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -9 + x 8*\-9 + x / 8*\/ -9 + x 4*\-9 + x / 4*\/ -9 + x 8*\-9 + x / | | 8* / -1 + -- | | \/ 9 | < dx | | 2 4 6 4 2 | | 81 27 81*x 5*x x 27*x 81*x | | - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ ________ 3/2 ________ 3/2 3/2 ________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 9 - x / x 8*\9 - x / 4*\/ 9 - x 4*\9 - x / 8*\9 - x / 8*\/ 9 - x | | 8* / 1 - -- | \ \/ 9 | / 0
$$\int\limits_{0}^{3} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i}{8 \sqrt{x^{2} - 9}} - \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81}{8 \sqrt{9 - x^{2}}} + \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
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3 / | | / 2 2 6 4 4 2 | | 27*I 81*I 81*I*x 81*I*x I*x 5*I*x 27*I*x x | |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 9 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -9 + x 8*\-9 + x / 8*\/ -9 + x 4*\-9 + x / 4*\/ -9 + x 8*\-9 + x / | | 8* / -1 + -- | | \/ 9 | < dx | | 2 4 6 4 2 | | 81 27 81*x 5*x x 27*x 81*x | | - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ ________ 3/2 ________ 3/2 3/2 ________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 9 - x / x 8*\9 - x / 4*\/ 9 - x 4*\9 - x / 8*\9 - x / 8*\/ 9 - x | | 8* / 1 - -- | \ \/ 9 | / 0
$$\int\limits_{0}^{3} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i}{8 \sqrt{x^{2} - 9}} - \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81}{8 \sqrt{9 - x^{2}}} + \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-27*i/(8*sqrt(-1 + x^2/9)) + 81*i/(8*sqrt(-9 + x^2)) - 81*i*x^2/(8*(-9 + x^2)^(3/2)) - 81*i*x^2/(8*sqrt(-9 + x^2)) - i*x^6/(4*(-9 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-9 + x^2)) + 27*i*x^4/(8*(-9 + x^2)^(3/2)), x^2/9 > 1), (-81/(8*sqrt(9 - x^2)) + 27/(8*sqrt(1 - x^2/9)) - 81*x^2/(8*(9 - x^2)^(3/2)) - 5*x^4/(4*sqrt(9 - x^2)) - x^6/(4*(9 - x^2)^(3/2)) + 27*x^4/(8*(9 - x^2)^(3/2)) + 81*x^2/(8*sqrt(9 - x^2)), True)), (x, 0, 3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.