Sr Examen
Integral de y*cos(2*x*y) dy
Solución
Ha introducido
[src]
pi / | | y*cos(2*x*y) dy | / pi -- 2
$$\int\limits_{\frac{\pi}{2}}^{\pi} y \cos{\left(2 x y \right)}\, dy$$
Integral(y*cos((2*x)*y), (y, pi/2, pi))
Respuesta (Indefinida)
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// 2 \
|| y |
|| -- for x = 0|
|| 2 |
/ || | // y for x = 0\
| ||/-cos(2*x*y) | || |
| y*cos(2*x*y) dy = C - |<|------------ for 2*x != 0 | + y*|
$$\int y \cos{\left(2 x y \right)}\, dy = C + y \left(\begin{cases} y & \text{for}\: x = 0 \\\frac{\sin{\left(2 x y \right)}}{2 x} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{y^{2}}{2} & \text{for}\: x = 0 \\\frac{\begin{cases} - \frac{\cos{\left(2 x y \right)}}{2 x} & \text{for}\: 2 x \neq 0 \\0 & \text{otherwise} \end{cases}}{2 x} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ cos(pi*x) cos(2*pi*x) pi*sin(2*pi*x) pi*sin(pi*x) |- --------- + ----------- + -------------- - ------------ for And(x > -oo, x < oo, x != 0) | 2 2 2*x 4*x | 4*x 4*x < | 2 | 3*pi | ----- otherwise \ 8
$$\begin{cases} - \frac{\pi \sin{\left(\pi x \right)}}{4 x} + \frac{\pi \sin{\left(2 \pi x \right)}}{2 x} - \frac{\cos{\left(\pi x \right)}}{4 x^{2}} + \frac{\cos{\left(2 \pi x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{3 \pi^{2}}{8} & \text{otherwise} \end{cases}$$
=
=
/ cos(pi*x) cos(2*pi*x) pi*sin(2*pi*x) pi*sin(pi*x) |- --------- + ----------- + -------------- - ------------ for And(x > -oo, x < oo, x != 0) | 2 2 2*x 4*x | 4*x 4*x < | 2 | 3*pi | ----- otherwise \ 8
$$\begin{cases} - \frac{\pi \sin{\left(\pi x \right)}}{4 x} + \frac{\pi \sin{\left(2 \pi x \right)}}{2 x} - \frac{\cos{\left(\pi x \right)}}{4 x^{2}} + \frac{\cos{\left(2 \pi x \right)}}{4 x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\\frac{3 \pi^{2}}{8} & \text{otherwise} \end{cases}$$
Piecewise((-cos(pi*x)/(4*x^2) + cos(2*pi*x)/(4*x^2) + pi*sin(2*pi*x)/(2*x) - pi*sin(pi*x)/(4*x), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (3*pi^2/8, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.