Sr Examen
Integral de x^2*(8-2x^2)^0,5 dx
Solución
Ha introducido
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2 / | | __________ | 2 / 2 | x *\/ 8 - 2*x dx | / ___ \/ 2
$$\int\limits_{\sqrt{2}}^{2} x^{2} \sqrt{8 - 2 x^{2}}\, dx$$
Integral(x^2*sqrt(8 - 2*x^2), (x, sqrt(2), 2))
Respuesta (Indefinida)
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| // ________ / 2\ \
| __________ || / 2 | x | |
| 2 / 2 ___ || x*\/ 4 - x *|1 - --| |
| x *\/ 8 - 2*x dx = C + \/ 2 *|< /x\ \ 2 / |
| ||2*asin|-| - ---------------------- for And(x > -2, x < 2)|
/ || \2/ 2 |
\\ /
$$\int x^{2} \sqrt{8 - 2 x^{2}}\, dx = C + \sqrt{2} \left(\begin{cases} - \frac{x \left(1 - \frac{x^{2}}{2}\right) \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)} & \text{for}\: x > -2 \wedge x < 2 \end{cases}\right)$$
Respuesta
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2 / | | / ___ ___ ___ 2 ___ 2 ___ 6 ___ 4 ___ 4 2 | | I*\/ 2 2*I*\/ 2 2*I*\/ 2 *x 9*I*\/ 2 *x I*\/ 2 *x 3*I*\/ 2 *x 5*I*\/ 2 *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 | | \/ 2 2*\/ 2 2*\/ 2 *x 5*\/ 2 *x \/ 2 *x 3*\/ 2 *x 9*\/ 2 *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 | / ___ \/ 2
$$\int\limits_{\sqrt{2}}^{2} \begin{cases} - \frac{\sqrt{2} i x^{6}}{4 \left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{5 \sqrt{2} i x^{4}}{4 \sqrt{x^{2} - 4}} + \frac{3 \sqrt{2} i x^{4}}{2 \left(x^{2} - 4\right)^{\frac{3}{2}}} - \frac{9 \sqrt{2} i x^{2}}{2 \sqrt{x^{2} - 4}} - \frac{2 \sqrt{2} i x^{2}}{\left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{2 \sqrt{2} i}{\sqrt{x^{2} - 4}} - \frac{\sqrt{2} i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\- \frac{\sqrt{2} x^{6}}{4 \left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 \sqrt{2} x^{4}}{4 \sqrt{4 - x^{2}}} + \frac{3 \sqrt{2} x^{4}}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 \sqrt{2} x^{2}}{2 \sqrt{4 - x^{2}}} - \frac{2 \sqrt{2} x^{2}}{\left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{2 \sqrt{2}}{\sqrt{4 - x^{2}}} + \frac{\sqrt{2}}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$
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2 / | | / ___ ___ ___ 2 ___ 2 ___ 6 ___ 4 ___ 4 2 | | I*\/ 2 2*I*\/ 2 2*I*\/ 2 *x 9*I*\/ 2 *x I*\/ 2 *x 3*I*\/ 2 *x 5*I*\/ 2 *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 | | \/ 2 2*\/ 2 2*\/ 2 *x 5*\/ 2 *x \/ 2 *x 3*\/ 2 *x 9*\/ 2 *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 | / ___ \/ 2
$$\int\limits_{\sqrt{2}}^{2} \begin{cases} - \frac{\sqrt{2} i x^{6}}{4 \left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{5 \sqrt{2} i x^{4}}{4 \sqrt{x^{2} - 4}} + \frac{3 \sqrt{2} i x^{4}}{2 \left(x^{2} - 4\right)^{\frac{3}{2}}} - \frac{9 \sqrt{2} i x^{2}}{2 \sqrt{x^{2} - 4}} - \frac{2 \sqrt{2} i x^{2}}{\left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{2 \sqrt{2} i}{\sqrt{x^{2} - 4}} - \frac{\sqrt{2} i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\- \frac{\sqrt{2} x^{6}}{4 \left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 \sqrt{2} x^{4}}{4 \sqrt{4 - x^{2}}} + \frac{3 \sqrt{2} x^{4}}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 \sqrt{2} x^{2}}{2 \sqrt{4 - x^{2}}} - \frac{2 \sqrt{2} x^{2}}{\left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{2 \sqrt{2}}{\sqrt{4 - x^{2}}} + \frac{\sqrt{2}}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*sqrt(2)/sqrt(-1 + x^2/4) + 2*i*sqrt(2)/sqrt(-4 + x^2) - 2*i*sqrt(2)*x^2/(-4 + x^2)^(3/2) - 9*i*sqrt(2)*x^2/(2*sqrt(-4 + x^2)) - i*sqrt(2)*x^6/(4*(-4 + x^2)^(3/2)) + 3*i*sqrt(2)*x^4/(2*(-4 + x^2)^(3/2)) + 5*i*sqrt(2)*x^4/(4*sqrt(-4 + x^2)), x^2/4 > 1), (sqrt(2)/sqrt(1 - x^2/4) - 2*sqrt(2)/sqrt(4 - x^2) - 2*sqrt(2)*x^2/(4 - x^2)^(3/2) - 5*sqrt(2)*x^4/(4*sqrt(4 - x^2)) - sqrt(2)*x^6/(4*(4 - x^2)^(3/2)) + 3*sqrt(2)*x^4/(2*(4 - x^2)^(3/2)) + 9*sqrt(2)*x^2/(2*sqrt(4 - x^2)), True)), (x, sqrt(2), 2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.