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