Sr Examen
Integral de x^3/sqrt(a^2*t^2-x^2) dx
Solución
Ha introducido
[src]
a*t / | | 3 | x | --------------- dx | ____________ | / 2 2 2 | \/ a *t - x | / 0
$$\int\limits_{0}^{a t} \frac{x^{3}}{\sqrt{a^{2} t^{2} - x^{2}}}\, dx$$
Integral(x^3/sqrt(a^2*t^2 - x^2), (x, 0, a*t))
Respuesta (Indefinida)
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// ____________ ____________ \
|| / 2 / 2 |
|| 3 3 / x 2 / x |
|| 2*I*a *t * / -1 + ----- I*a*t*x * / -1 + ----- |
|| / 2 2 / 2 2 | 2 | |
/ || \/ a *t \/ a *t | x | |
| ||- ---------------------------- - --------------------------- for |-----| > 1|
| 3 || 3 3 | 2 2| |
| x || |a *t | |
| --------------- dx = C + |< |
| ____________ || ___________ ___________ |
| / 2 2 2 || / 2 / 2 |
| \/ a *t - x || 3 3 / x 2 / x |
| || 2*a *t * / 1 - ----- a*t*x * / 1 - ----- |
/ || / 2 2 / 2 2 |
|| \/ a *t \/ a *t |
|| - ------------------------- - ------------------------ otherwise |
|| 3 3 |
\\ /
$$\int \frac{x^{3}}{\sqrt{a^{2} t^{2} - x^{2}}}\, dx = C + \begin{cases} - \frac{2 i a^{3} t^{3} \sqrt{-1 + \frac{x^{2}}{a^{2} t^{2}}}}{3} - \frac{i a t x^{2} \sqrt{-1 + \frac{x^{2}}{a^{2} t^{2}}}}{3} & \text{for}\: \left|{\frac{x^{2}}{a^{2} t^{2}}}\right| > 1 \\- \frac{2 a^{3} t^{3} \sqrt{1 - \frac{x^{2}}{a^{2} t^{2}}}}{3} - \frac{a t x^{2} \sqrt{1 - \frac{x^{2}}{a^{2} t^{2}}}}{3} & \text{otherwise} \end{cases}$$
Respuesta
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_______
2 2 / 2 2
2*a *t *\/ a *t
------------------
3
$$\frac{2 a^{2} t^{2} \sqrt{a^{2} t^{2}}}{3}$$
=
=
_______
2 2 / 2 2
2*a *t *\/ a *t
------------------
3
$$\frac{2 a^{2} t^{2} \sqrt{a^{2} t^{2}}}{3}$$
2*a^2*t^2*sqrt(a^2*t^2)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.