Sr Examen
Integral de (1-x)(cos(n*pi*x/2)) dx
Solución
Ha introducido
[src]
6 / | | /n*pi*x\ | (1 - x)*cos|------| dx | \ 2 / | / 0
$$\int\limits_{0}^{6} \left(1 - x\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx$$
Integral((1 - x)*cos(((n*pi)*x)/2), (x, 0, 6))
Respuesta (Indefinida)
[src]
// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ // x for n = 0\ // x for n = 0\ || |
| || | || | || // /pi*n*x\ \ |
| /n*pi*x\ || /pi*n*x\ | || /pi*n*x\ | || ||-2*cos|------| | |
| (1 - x)*cos|------| dx = C - x*|<2*sin|------| | + |<2*sin|------| | + |< || \ 2 / pi*n | |
| \ 2 / || \ 2 / | || \ 2 / | ||2*|<-------------- for ---- != 0| |
| ||------------- otherwise| ||------------- otherwise| || || pi*n 2 | |
/ \\ pi*n / \\ pi*n / || || | |
|| \\ 0 otherwise / |
||---------------------------------- otherwise|
|| pi*n |
\\ /
$$\int \left(1 - x\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C - x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + \begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases} + \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 4 10*sin(3*pi*n) 4*cos(3*pi*n) |------ - -------------- - ------------- for And(n > -oo, n < oo, n != 0) | 2 2 pi*n 2 2
$$\begin{cases} - \frac{10 \sin{\left(3 \pi n \right)}}{\pi n} - \frac{4 \cos{\left(3 \pi n \right)}}{\pi^{2} n^{2}} + \frac{4}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-12 & \text{otherwise} \end{cases}$$
=
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/ 4 10*sin(3*pi*n) 4*cos(3*pi*n) |------ - -------------- - ------------- for And(n > -oo, n < oo, n != 0) | 2 2 pi*n 2 2
$$\begin{cases} - \frac{10 \sin{\left(3 \pi n \right)}}{\pi n} - \frac{4 \cos{\left(3 \pi n \right)}}{\pi^{2} n^{2}} + \frac{4}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\-12 & \text{otherwise} \end{cases}$$
Piecewise((4/(pi^2*n^2) - 10*sin(3*pi*n)/(pi*n) - 4*cos(3*pi*n)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-12, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.