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