Sr Examen
Integral de (xy^2)^cos(z) dx
Solución
Ha introducido
[src]
1 / | | cos(z) | / 2\ | \x*y / dx | / 0
$$\int\limits_{0}^{1} \left(x y^{2}\right)^{\cos{\left(z \right)}}\, dx$$
Integral((x*y^2)^cos(z), (x, 0, 1))
Respuesta (Indefinida)
[src]
/// 1 + cos(z) \
|||/ 2\ |
|||\x*y / |
/ |||---------------- for cos(z) != -1 |
| ||< 1 + cos(z) |
| cos(z) ||| |
| / 2\ ||| / 2\ |
| \x*y / dx = C + |<| log\x*y / otherwise |
| ||\ 2 |
/ ||----------------------------------- for y != 0|
|| 2 |
|| y |
|| |
|| cos(z) |
\\ x*0 otherwise /
$$\int \left(x y^{2}\right)^{\cos{\left(z \right)}}\, dx = C + \begin{cases} \frac{\begin{cases} \frac{\left(x y^{2}\right)^{\cos{\left(z \right)} + 1}}{\cos{\left(z \right)} + 1} & \text{for}\: \cos{\left(z \right)} \neq -1 \\\log{\left(x y^{2} \right)} & \text{otherwise} \end{cases}}{y^{2}} & \text{for}\: y^{2} \neq 0 \\0^{\cos{\left(z \right)}} x & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 1 + cos(z) | / 2\ 1 + cos(z) | \y / 0 |--------------- - --------------- for Or(And(z >= 0, z < pi), And(z <= 2*pi, z > pi)) | 2 2 |y *(1 + cos(z)) y *(1 + cos(z)) < | / 2\ | /1 \ log\y / | oo*sign|--| + ------- otherwise | | 2| 2 | \y / y \
$$\begin{cases} - \frac{0^{\cos{\left(z \right)} + 1}}{y^{2} \left(\cos{\left(z \right)} + 1\right)} + \frac{\left(y^{2}\right)^{\cos{\left(z \right)} + 1}}{y^{2} \left(\cos{\left(z \right)} + 1\right)} & \text{for}\: \left(z \geq 0 \wedge z < \pi\right) \vee \left(z \leq 2 \pi \wedge z > \pi\right) \\\infty \operatorname{sign}{\left(\frac{1}{y^{2}} \right)} + \frac{\log{\left(y^{2} \right)}}{y^{2}} & \text{otherwise} \end{cases}$$
=
=
/ 1 + cos(z) | / 2\ 1 + cos(z) | \y / 0 |--------------- - --------------- for Or(And(z >= 0, z < pi), And(z <= 2*pi, z > pi)) | 2 2 |y *(1 + cos(z)) y *(1 + cos(z)) < | / 2\ | /1 \ log\y / | oo*sign|--| + ------- otherwise | | 2| 2 | \y / y \
$$\begin{cases} - \frac{0^{\cos{\left(z \right)} + 1}}{y^{2} \left(\cos{\left(z \right)} + 1\right)} + \frac{\left(y^{2}\right)^{\cos{\left(z \right)} + 1}}{y^{2} \left(\cos{\left(z \right)} + 1\right)} & \text{for}\: \left(z \geq 0 \wedge z < \pi\right) \vee \left(z \leq 2 \pi \wedge z > \pi\right) \\\infty \operatorname{sign}{\left(\frac{1}{y^{2}} \right)} + \frac{\log{\left(y^{2} \right)}}{y^{2}} & \text{otherwise} \end{cases}$$
Piecewise(((y^2)^(1 + cos(z))/(y^2*(1 + cos(z))) - 0^(1 + cos(z))/(y^2*(1 + cos(z))), ((z >= 0)∧(z < pi))∨((z > pi)∧(z <= 2*pi))), (oo*sign(y^(-2)) + log(y^2)/y^2, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.