Sr Examen
Integral de x*(x-2)*cos(pi*n*x/2) dx
Solución
Ha introducido
[src]
2 / | | /pi*n*x\ | x*(x - 2)*cos|------| dx | \ 2 / | / 0
$$\int\limits_{0}^{2} x \left(x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx$$
Integral((x*(x - 2))*cos(((pi*n)*x)/2), (x, 0, 2))
Respuesta (Indefinida)
[src]
// 3 \
|| x | // 2 \
|| -- for n = 0| || x |
|| 3 | || -- for n = 0|
|| | || 2 |
/ || // /pi*n*x\ /pi*n*x\ \ | || | // x for n = 0\ // x for n = 0\
| || ||4*sin|------| 2*x*cos|------| | | || // /pi*n*x\ \ | || | || |
| /pi*n*x\ || || \ 2 / \ 2 / | | || ||-2*cos|------| | | 2 || /pi*n*x\ | || /pi*n*x\ |
| x*(x - 2)*cos|------| dx = C - 2*|< ||------------- - --------------- for n != 0| | + 2*|< || \ 2 / pi*n | | + x *|<2*sin|------| | - 2*x*|<2*sin|------| |
| \ 2 / ||2*|< 2 2 pi*n | | ||2*|<-------------- for ---- != 0| | || \ 2 / | || \ 2 / |
| || || pi *n | | || || pi*n 2 | | ||------------- otherwise| ||------------- otherwise|
/ || || | | || || | | \\ pi*n / \\ pi*n /
|| || 0 otherwise | | || \\ 0 otherwise / |
|| \\ / | ||---------------------------------- otherwise|
||------------------------------------------------ otherwise| || pi*n |
|| pi*n | \\ /
\\ /
$$\int x \left(x - 2\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C + x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 2 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + 2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 x \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} + \frac{4 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi^{2} n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ 8 16*sin(pi*n) 8*cos(pi*n) |------ - ------------ + ----------- for And(n > -oo, n < oo, n != 0) | 2 2 3 3 2 2
$$\begin{cases} \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{8}{\pi^{2} n^{2}} - \frac{16 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4}{3} & \text{otherwise} \end{cases}$$
=
=
/ 8 16*sin(pi*n) 8*cos(pi*n) |------ - ------------ + ----------- for And(n > -oo, n < oo, n != 0) | 2 2 3 3 2 2
$$\begin{cases} \frac{8 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} + \frac{8}{\pi^{2} n^{2}} - \frac{16 \sin{\left(\pi n \right)}}{\pi^{3} n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4}{3} & \text{otherwise} \end{cases}$$
Piecewise((8/(pi^2*n^2) - 16*sin(pi*n)/(pi^3*n^3) + 8*cos(pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-4/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.