Sr Examen
Integral de y^2*cos(pi*x*y) dy
Solución
Ha introducido
[src]
1 / | | 2 | y *cos(pi*x*y) dy | / 0
$$\int\limits_{0}^{1} y^{2} \cos{\left(y \pi x \right)}\, dy$$
Integral(y^2*cos((pi*x)*y), (y, 0, 1))
Respuesta (Indefinida)
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// 3 \
|| y |
|| -- for x = 0|
|| 3 |
/ || |
| ||/sin(pi*x*y) y*cos(pi*x*y) | // y for x = 0\
| 2 |||----------- - ------------- for x != 0 | 2 || |
| y *cos(pi*x*y) dy = C - 2*|<| 2 2 pi*x | + y *|
$$\int y^{2} \cos{\left(y \pi x \right)}\, dy = C + y^{2} \left(\begin{cases} y & \text{for}\: x = 0 \\\frac{\sin{\left(\pi x y \right)}}{\pi x} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{y^{3}}{3} & \text{for}\: x = 0 \\\frac{\begin{cases} - \frac{y \cos{\left(\pi x y \right)}}{\pi x} + \frac{\sin{\left(\pi x y \right)}}{\pi^{2} x^{2}} & \text{for}\: x \neq 0 \\0 & \text{otherwise} \end{cases}}{\pi x} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/sin(pi*x) 2*sin(pi*x) 2*cos(pi*x) |--------- - ----------- + ----------- for And(x > -oo, x < oo, x != 0) | pi*x 3 3 2 2 < pi *x pi *x | | 1/3 otherwise \
$$\begin{cases} \frac{\sin{\left(\pi x \right)}}{\pi x} + \frac{2 \cos{\left(\pi x \right)}}{\pi^{2} x^{2}} - \frac{2 \sin{\left(\pi x \right)}}{\pi^{3} x^{3}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
=
=
/sin(pi*x) 2*sin(pi*x) 2*cos(pi*x) |--------- - ----------- + ----------- for And(x > -oo, x < oo, x != 0) | pi*x 3 3 2 2 < pi *x pi *x | | 1/3 otherwise \
$$\begin{cases} \frac{\sin{\left(\pi x \right)}}{\pi x} + \frac{2 \cos{\left(\pi x \right)}}{\pi^{2} x^{2}} - \frac{2 \sin{\left(\pi x \right)}}{\pi^{3} x^{3}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
Piecewise((sin(pi*x)/(pi*x) - 2*sin(pi*x)/(pi^3*x^3) + 2*cos(pi*x)/(pi^2*x^2), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (1/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.