Sr Examen
Integral de x^2*(sqrt(7-x^2)) dx
Solución
Ha introducido
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1 / | | ________ | 2 / 2 | x *\/ 7 - x dx | / 0
$$\int\limits_{0}^{1} x^{2} \sqrt{7 - x^{2}}\, dx$$
Integral(x^2*sqrt(7 - 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=sqrt(7)*sin(_theta), rewritten=49/8 - 49*cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=49/8, context=49/8, symbol=_theta), ConstantTimesRule(constant=-49/8, 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=-49*cos(4*_theta)/8, symbol=_theta)], context=49/8 - 49*cos(4*_theta)/8, symbol=_theta), restriction=(x < sqrt(7)) & (x > -sqrt(7)), context=x**2*sqrt(7 - x**2), symbol=x)
Respuesta:
Respuesta (Indefinida)
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/
| // / ___\ \
| ________ || |x*\/ 7 | ________ |
| 2 / 2 ||49*asin|-------| / 2 / 2\ |
| x *\/ 7 - x dx = C + |< \ 7 / x*\/ 7 - x *\7 - 2*x / / ___ ___\|
| ||---------------- - ------------------------ for And\x > -\/ 7 , x < \/ 7 /|
/ || 8 8 |
\\ /
$$\int x^{2} \sqrt{7 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(7 - 2 x^{2}\right) \sqrt{7 - x^{2}}}{8} + \frac{49 \operatorname{asin}{\left(\frac{\sqrt{7} x}{7} \right)}}{8} & \text{for}\: x > - \sqrt{7} \wedge x < \sqrt{7} \end{cases}$$
Gráfica
Respuesta
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1 / | | / 2 2 ___ 6 4 4 2 | | 49*I 63*I*x 49*I*x 7*I*\/ 7 I*x 5*I*x 21*I*x x | |-------------- - -------------- - -------------- - ---------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 7 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | |8*\/ -7 + x 8*\/ -7 + x 8*\-7 + x / / x 4*\-7 + x / 4*\/ -7 + x 8*\-7 + x / | | 8* / -1 + -- | | \/ 7 | < dx | | 2 4 6 ___ 4 2 | | 49 49*x 5*x x 7*\/ 7 21*x 63*x | | - ------------- - ------------- - ------------- - ------------- + --------------- + ------------- + ------------- otherwise | | ________ 3/2 ________ 3/2 ________ 3/2 ________ | | / 2 / 2\ / 2 / 2\ / 2 / 2\ / 2 | | 8*\/ 7 - x 8*\7 - x / 4*\/ 7 - x 4*\7 - x / / x 8*\7 - x / 8*\/ 7 - x | | 8* / 1 - -- | \ \/ 7 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 7\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 7}} + \frac{21 i x^{4}}{8 \left(x^{2} - 7\right)^{\frac{3}{2}}} - \frac{63 i x^{2}}{8 \sqrt{x^{2} - 7}} - \frac{49 i x^{2}}{8 \left(x^{2} - 7\right)^{\frac{3}{2}}} + \frac{49 i}{8 \sqrt{x^{2} - 7}} - \frac{7 \sqrt{7} i}{8 \sqrt{\frac{x^{2}}{7} - 1}} & \text{for}\: \frac{x^{2}}{7} > 1 \\- \frac{x^{6}}{4 \left(7 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{7 - x^{2}}} + \frac{21 x^{4}}{8 \left(7 - x^{2}\right)^{\frac{3}{2}}} + \frac{63 x^{2}}{8 \sqrt{7 - x^{2}}} - \frac{49 x^{2}}{8 \left(7 - x^{2}\right)^{\frac{3}{2}}} - \frac{49}{8 \sqrt{7 - x^{2}}} + \frac{7 \sqrt{7}}{8 \sqrt{1 - \frac{x^{2}}{7}}} & \text{otherwise} \end{cases}\, dx$$
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1 / | | / 2 2 ___ 6 4 4 2 | | 49*I 63*I*x 49*I*x 7*I*\/ 7 I*x 5*I*x 21*I*x x | |-------------- - -------------- - -------------- - ---------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 7 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | |8*\/ -7 + x 8*\/ -7 + x 8*\-7 + x / / x 4*\-7 + x / 4*\/ -7 + x 8*\-7 + x / | | 8* / -1 + -- | | \/ 7 | < dx | | 2 4 6 ___ 4 2 | | 49 49*x 5*x x 7*\/ 7 21*x 63*x | | - ------------- - ------------- - ------------- - ------------- + --------------- + ------------- + ------------- otherwise | | ________ 3/2 ________ 3/2 ________ 3/2 ________ | | / 2 / 2\ / 2 / 2\ / 2 / 2\ / 2 | | 8*\/ 7 - x 8*\7 - x / 4*\/ 7 - x 4*\7 - x / / x 8*\7 - x / 8*\/ 7 - x | | 8* / 1 - -- | \ \/ 7 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 7\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 7}} + \frac{21 i x^{4}}{8 \left(x^{2} - 7\right)^{\frac{3}{2}}} - \frac{63 i x^{2}}{8 \sqrt{x^{2} - 7}} - \frac{49 i x^{2}}{8 \left(x^{2} - 7\right)^{\frac{3}{2}}} + \frac{49 i}{8 \sqrt{x^{2} - 7}} - \frac{7 \sqrt{7} i}{8 \sqrt{\frac{x^{2}}{7} - 1}} & \text{for}\: \frac{x^{2}}{7} > 1 \\- \frac{x^{6}}{4 \left(7 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{7 - x^{2}}} + \frac{21 x^{4}}{8 \left(7 - x^{2}\right)^{\frac{3}{2}}} + \frac{63 x^{2}}{8 \sqrt{7 - x^{2}}} - \frac{49 x^{2}}{8 \left(7 - x^{2}\right)^{\frac{3}{2}}} - \frac{49}{8 \sqrt{7 - x^{2}}} + \frac{7 \sqrt{7}}{8 \sqrt{1 - \frac{x^{2}}{7}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((49*i/(8*sqrt(-7 + x^2)) - 63*i*x^2/(8*sqrt(-7 + x^2)) - 49*i*x^2/(8*(-7 + x^2)^(3/2)) - 7*i*sqrt(7)/(8*sqrt(-1 + x^2/7)) - i*x^6/(4*(-7 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-7 + x^2)) + 21*i*x^4/(8*(-7 + x^2)^(3/2)), x^2/7 > 1), (-49/(8*sqrt(7 - x^2)) - 49*x^2/(8*(7 - x^2)^(3/2)) - 5*x^4/(4*sqrt(7 - x^2)) - x^6/(4*(7 - x^2)^(3/2)) + 7*sqrt(7)/(8*sqrt(1 - x^2/7)) + 21*x^4/(8*(7 - x^2)^(3/2)) + 63*x^2/(8*sqrt(7 - x^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.