Sr Examen
Integral de x^2sqrt(x^2-9) dx
Solución
Ha introducido
[src]
1 / | | ________ | 2 / 2 | x *\/ x - 9 dx | / 0
$$\int\limits_{0}^{1} x^{2} \sqrt{x^{2} - 9}\, dx$$
Integral(x^2*sqrt(x^2 - 9), (x, 0, 1))
Respuesta (Indefinida)
[src]
// /x\ \
|| 81*acosh|-| 3 5 | 2| |
|| \3/ 27*x x 81*x |x | |
/ ||- ----------- - -------------- + -------------- + -------------- for ---- > 1|
| || 8 _________ _________ _________ 9 |
| ________ || / 2 / 2 / 2 |
| 2 / 2 || 8*\/ -9 + x 4*\/ -9 + x 8*\/ -9 + x |
| x *\/ x - 9 dx = C + |< |
| || /x\ |
/ || 81*I*asin|-| 5 3 |
|| \3/ 81*I*x I*x 27*I*x |
|| ------------ - ------------- - ------------- + ------------- otherwise |
|| 8 ________ ________ ________ |
|| / 2 / 2 / 2 |
\\ 8*\/ 9 - x 4*\/ 9 - x 8*\/ 9 - x /
$$\int x^{2} \sqrt{x^{2} - 9}\, dx = C + \begin{cases} \frac{x^{5}}{4 \sqrt{x^{2} - 9}} - \frac{27 x^{3}}{8 \sqrt{x^{2} - 9}} + \frac{81 x}{8 \sqrt{x^{2} - 9}} - \frac{81 \operatorname{acosh}{\left(\frac{x}{3} \right)}}{8} & \text{for}\: \frac{\left|{x^{2}}\right|}{9} > 1 \\- \frac{i x^{5}}{4 \sqrt{9 - x^{2}}} + \frac{27 i x^{3}}{8 \sqrt{9 - x^{2}}} - \frac{81 i x}{8 \sqrt{9 - x^{2}}} + \frac{81 i \operatorname{asin}{\left(\frac{x}{3} \right)}}{8} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
[src]
1 / | | / 2 2 6 4 4 2 | | 27 81 81*x 81*x x 5*x 27*x x | |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 9 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -9 + x 8*\-9 + x / 8*\/ -9 + x 4*\-9 + x / 4*\/ -9 + x 8*\-9 + x / | | 8* / -1 + -- | | \/ 9 | < dx | | 2 4 6 4 2 | | 81*I 27*I 81*I*x 5*I*x I*x 27*I*x 81*I*x | | - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ ________ 3/2 ________ 3/2 3/2 ________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 9 - x / x 8*\9 - x / 4*\/ 9 - x 4*\9 - x / 8*\9 - x / 8*\/ 9 - x | | 8* / 1 - -- | \ \/ 9 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81}{8 \sqrt{x^{2} - 9}} - \frac{27}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{i x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 i x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 i x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 i x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81 i}{8 \sqrt{9 - x^{2}}} + \frac{27 i}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
=
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1 / | | / 2 2 6 4 4 2 | | 27 81 81*x 81*x x 5*x 27*x x | |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + -------------- for -- > 1 | | _________ _________ 3/2 _________ 3/2 _________ 3/2 9 | | / 2 / 2 / 2\ / 2 / 2\ / 2 / 2\ | | / x 8*\/ -9 + x 8*\-9 + x / 8*\/ -9 + x 4*\-9 + x / 4*\/ -9 + x 8*\-9 + x / | | 8* / -1 + -- | | \/ 9 | < dx | | 2 4 6 4 2 | | 81*I 27*I 81*I*x 5*I*x I*x 27*I*x 81*I*x | | - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ ________ 3/2 ________ 3/2 3/2 ________ | | / 2 / 2 / 2\ / 2 / 2\ / 2\ / 2 | | 8*\/ 9 - x / x 8*\9 - x / 4*\/ 9 - x 4*\9 - x / 8*\9 - x / 8*\/ 9 - x | | 8* / 1 - -- | \ \/ 9 | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81}{8 \sqrt{x^{2} - 9}} - \frac{27}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{i x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 i x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 i x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 i x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81 i}{8 \sqrt{9 - x^{2}}} + \frac{27 i}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-27/(8*sqrt(-1 + x^2/9)) + 81/(8*sqrt(-9 + x^2)) - 81*x^2/(8*(-9 + x^2)^(3/2)) - 81*x^2/(8*sqrt(-9 + x^2)) - x^6/(4*(-9 + x^2)^(3/2)) + 5*x^4/(4*sqrt(-9 + x^2)) + 27*x^4/(8*(-9 + x^2)^(3/2)), x^2/9 > 1), (-81*i/(8*sqrt(9 - x^2)) + 27*i/(8*sqrt(1 - x^2/9)) - 81*i*x^2/(8*(9 - x^2)^(3/2)) - 5*i*x^4/(4*sqrt(9 - x^2)) - i*x^6/(4*(9 - x^2)^(3/2)) + 27*i*x^4/(8*(9 - x^2)^(3/2)) + 81*i*x^2/(8*sqrt(9 - x^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.