Sr Examen
Integral de x^2*sqrt(49-x^2) dx
Solución
Ha introducido
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1 / | | _________ | 2 / 2 | x *\/ 49 - x dx | / 0
$$\int\limits_{0}^{1} x^{2} \sqrt{49 - x^{2}}\, dx$$
Integral(x^2*sqrt(49 - 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=7*sin(_theta), rewritten=2401/8 - 2401*cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=2401/8, context=2401/8, symbol=_theta), ConstantTimesRule(constant=-2401/8, 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=-2401*cos(4*_theta)/8, symbol=_theta)], context=2401/8 - 2401*cos(4*_theta)/8, symbol=_theta), restriction=(x > -7) & (x < 7), context=x**2*sqrt(49 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | _________ // /x\ _________ \ | 2 / 2 ||2401*asin|-| / 2 / 2\ | | x *\/ 49 - x dx = C + |< \7/ x*\/ 49 - x *\49 - 2*x / | | ||------------ - -------------------------- for And(x > -7, x < 7)| / \\ 8 8 /
$$\int x^{2} \sqrt{49 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(49 - 2 x^{2}\right) \sqrt{49 - x^{2}}}{8} + \frac{2401 \operatorname{asin}{\left(\frac{x}{7} \right)}}{8} & \text{for}\: x > -7 \wedge x < 7 \end{cases}$$
Gráfica
Respuesta
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1 / | | / 2 2 6 4 4 2 | | 343*I 2401*I 2401*I*x 441*I*x I*x 5*I*x 147*I*x x | |- ---------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 __________ 3/2 49 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -49 + x 8*\-49 + x / 8*\/ -49 + x 4*\-49 + x / 4*\/ -49 + x 8*\-49 + x / | | 8* / -1 + -- | | \/ 49 | < dx | | 2 4 6 4 2 | | 2401 343 2401*x 5*x x 147*x 441*x | | - -------------- + --------------- - -------------- - -------------- - -------------- + -------------- + -------------- otherwise | | _________ ________ 3/2 _________ 3/2 3/2 _________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 49 - x / x 8*\49 - x / 4*\/ 49 - x 4*\49 - x / 8*\49 - x / 8*\/ 49 - x | | 8* / 1 - -- | \ \/ 49 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 49\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 49}} + \frac{147 i x^{4}}{8 \left(x^{2} - 49\right)^{\frac{3}{2}}} - \frac{441 i x^{2}}{8 \sqrt{x^{2} - 49}} - \frac{2401 i x^{2}}{8 \left(x^{2} - 49\right)^{\frac{3}{2}}} + \frac{2401 i}{8 \sqrt{x^{2} - 49}} - \frac{343 i}{8 \sqrt{\frac{x^{2}}{49} - 1}} & \text{for}\: \frac{x^{2}}{49} > 1 \\- \frac{x^{6}}{4 \left(49 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{49 - x^{2}}} + \frac{147 x^{4}}{8 \left(49 - x^{2}\right)^{\frac{3}{2}}} + \frac{441 x^{2}}{8 \sqrt{49 - x^{2}}} - \frac{2401 x^{2}}{8 \left(49 - x^{2}\right)^{\frac{3}{2}}} - \frac{2401}{8 \sqrt{49 - x^{2}}} + \frac{343}{8 \sqrt{1 - \frac{x^{2}}{49}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / 2 2 6 4 4 2 | | 343*I 2401*I 2401*I*x 441*I*x I*x 5*I*x 147*I*x x | |- ---------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 __________ 3/2 49 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -49 + x 8*\-49 + x / 8*\/ -49 + x 4*\-49 + x / 4*\/ -49 + x 8*\-49 + x / | | 8* / -1 + -- | | \/ 49 | < dx | | 2 4 6 4 2 | | 2401 343 2401*x 5*x x 147*x 441*x | | - -------------- + --------------- - -------------- - -------------- - -------------- + -------------- + -------------- otherwise | | _________ ________ 3/2 _________ 3/2 3/2 _________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 49 - x / x 8*\49 - x / 4*\/ 49 - x 4*\49 - x / 8*\49 - x / 8*\/ 49 - x | | 8* / 1 - -- | \ \/ 49 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 49\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 49}} + \frac{147 i x^{4}}{8 \left(x^{2} - 49\right)^{\frac{3}{2}}} - \frac{441 i x^{2}}{8 \sqrt{x^{2} - 49}} - \frac{2401 i x^{2}}{8 \left(x^{2} - 49\right)^{\frac{3}{2}}} + \frac{2401 i}{8 \sqrt{x^{2} - 49}} - \frac{343 i}{8 \sqrt{\frac{x^{2}}{49} - 1}} & \text{for}\: \frac{x^{2}}{49} > 1 \\- \frac{x^{6}}{4 \left(49 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{49 - x^{2}}} + \frac{147 x^{4}}{8 \left(49 - x^{2}\right)^{\frac{3}{2}}} + \frac{441 x^{2}}{8 \sqrt{49 - x^{2}}} - \frac{2401 x^{2}}{8 \left(49 - x^{2}\right)^{\frac{3}{2}}} - \frac{2401}{8 \sqrt{49 - x^{2}}} + \frac{343}{8 \sqrt{1 - \frac{x^{2}}{49}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-343*i/(8*sqrt(-1 + x^2/49)) + 2401*i/(8*sqrt(-49 + x^2)) - 2401*i*x^2/(8*(-49 + x^2)^(3/2)) - 441*i*x^2/(8*sqrt(-49 + x^2)) - i*x^6/(4*(-49 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-49 + x^2)) + 147*i*x^4/(8*(-49 + x^2)^(3/2)), x^2/49 > 1), (-2401/(8*sqrt(49 - x^2)) + 343/(8*sqrt(1 - x^2/49)) - 2401*x^2/(8*(49 - x^2)^(3/2)) - 5*x^4/(4*sqrt(49 - x^2)) - x^6/(4*(49 - x^2)^(3/2)) + 147*x^4/(8*(49 - x^2)^(3/2)) + 441*x^2/(8*sqrt(49 - x^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.