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