Sr Examen
Integral de (x^2)*(sqrt(25-x^2)) dx
Solución
Ha introducido
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5 / | | _________ | 2 / 2 | x *\/ 25 - x dx | / 0
$$\int\limits_{0}^{5} x^{2} \sqrt{25 - x^{2}}\, dx$$
Integral(x^2*sqrt(25 - x^2), (x, 0, 5))
Solución detallada
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Ahora simplificar:
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Añadimos la constante de integración:
TrigSubstitutionRule(theta=_theta, func=5*sin(_theta), rewritten=625/8 - 625*cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=625/8, context=625/8, symbol=_theta), ConstantTimesRule(constant=-625/8, 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=-625*cos(4*_theta)/8, symbol=_theta)], context=625/8 - 625*cos(4*_theta)/8, symbol=_theta), restriction=(x > -5) & (x < 5), context=x**2*sqrt(25 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/ | | _________ // /x\ _________ \ | 2 / 2 ||625*asin|-| / 2 / 2\ | | x *\/ 25 - x dx = C + |< \5/ x*\/ 25 - x *\25 - 2*x / | | ||----------- - -------------------------- for And(x > -5, x < 5)| / \\ 8 8 /
$$\int x^{2} \sqrt{25 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(25 - 2 x^{2}\right) \sqrt{25 - x^{2}}}{8} + \frac{625 \operatorname{asin}{\left(\frac{x}{5} \right)}}{8} & \text{for}\: x > -5 \wedge x < 5 \end{cases}$$
Gráfica
Respuesta
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5 / | | / 2 2 6 4 4 2 | | 125*I 625*I 625*I*x 225*I*x I*x 5*I*x 75*I*x x | |- ---------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 __________ 3/2 25 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -25 + x 8*\-25 + x / 8*\/ -25 + x 4*\-25 + x / 4*\/ -25 + x 8*\-25 + x / | | 8* / -1 + -- | | \/ 25 | < dx | | 2 4 6 4 2 | | 625 125 625*x 5*x x 75*x 225*x | | - -------------- + --------------- - -------------- - -------------- - -------------- + -------------- + -------------- otherwise | | _________ ________ 3/2 _________ 3/2 3/2 _________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 25 - x / x 8*\25 - x / 4*\/ 25 - x 4*\25 - x / 8*\25 - x / 8*\/ 25 - x | | 8* / 1 - -- | \ \/ 25 | / 0
$$\int\limits_{0}^{5} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 25\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 25}} + \frac{75 i x^{4}}{8 \left(x^{2} - 25\right)^{\frac{3}{2}}} - \frac{225 i x^{2}}{8 \sqrt{x^{2} - 25}} - \frac{625 i x^{2}}{8 \left(x^{2} - 25\right)^{\frac{3}{2}}} + \frac{625 i}{8 \sqrt{x^{2} - 25}} - \frac{125 i}{8 \sqrt{\frac{x^{2}}{25} - 1}} & \text{for}\: \frac{x^{2}}{25} > 1 \\- \frac{x^{6}}{4 \left(25 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{25 - x^{2}}} + \frac{75 x^{4}}{8 \left(25 - x^{2}\right)^{\frac{3}{2}}} + \frac{225 x^{2}}{8 \sqrt{25 - x^{2}}} - \frac{625 x^{2}}{8 \left(25 - x^{2}\right)^{\frac{3}{2}}} - \frac{625}{8 \sqrt{25 - x^{2}}} + \frac{125}{8 \sqrt{1 - \frac{x^{2}}{25}}} & \text{otherwise} \end{cases}\, dx$$
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5 / | | / 2 2 6 4 4 2 | | 125*I 625*I 625*I*x 225*I*x I*x 5*I*x 75*I*x x | |- ---------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- for -- > 1 | | _________ __________ 3/2 __________ 3/2 __________ 3/2 25 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -25 + x 8*\-25 + x / 8*\/ -25 + x 4*\-25 + x / 4*\/ -25 + x 8*\-25 + x / | | 8* / -1 + -- | | \/ 25 | < dx | | 2 4 6 4 2 | | 625 125 625*x 5*x x 75*x 225*x | | - -------------- + --------------- - -------------- - -------------- - -------------- + -------------- + -------------- otherwise | | _________ ________ 3/2 _________ 3/2 3/2 _________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 25 - x / x 8*\25 - x / 4*\/ 25 - x 4*\25 - x / 8*\25 - x / 8*\/ 25 - x | | 8* / 1 - -- | \ \/ 25 | / 0
$$\int\limits_{0}^{5} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 25\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 25}} + \frac{75 i x^{4}}{8 \left(x^{2} - 25\right)^{\frac{3}{2}}} - \frac{225 i x^{2}}{8 \sqrt{x^{2} - 25}} - \frac{625 i x^{2}}{8 \left(x^{2} - 25\right)^{\frac{3}{2}}} + \frac{625 i}{8 \sqrt{x^{2} - 25}} - \frac{125 i}{8 \sqrt{\frac{x^{2}}{25} - 1}} & \text{for}\: \frac{x^{2}}{25} > 1 \\- \frac{x^{6}}{4 \left(25 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{25 - x^{2}}} + \frac{75 x^{4}}{8 \left(25 - x^{2}\right)^{\frac{3}{2}}} + \frac{225 x^{2}}{8 \sqrt{25 - x^{2}}} - \frac{625 x^{2}}{8 \left(25 - x^{2}\right)^{\frac{3}{2}}} - \frac{625}{8 \sqrt{25 - x^{2}}} + \frac{125}{8 \sqrt{1 - \frac{x^{2}}{25}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-125*i/(8*sqrt(-1 + x^2/25)) + 625*i/(8*sqrt(-25 + x^2)) - 625*i*x^2/(8*(-25 + x^2)^(3/2)) - 225*i*x^2/(8*sqrt(-25 + x^2)) - i*x^6/(4*(-25 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-25 + x^2)) + 75*i*x^4/(8*(-25 + x^2)^(3/2)), x^2/25 > 1), (-625/(8*sqrt(25 - x^2)) + 125/(8*sqrt(1 - x^2/25)) - 625*x^2/(8*(25 - x^2)^(3/2)) - 5*x^4/(4*sqrt(25 - x^2)) - x^6/(4*(25 - x^2)^(3/2)) + 75*x^4/(8*(25 - x^2)^(3/2)) + 225*x^2/(8*sqrt(25 - x^2)), True)), (x, 0, 5))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.