Sr Examen
Integral de (8-4x)*cos(n*pi*x/2) dx
Solución
Ha introducido
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2 / | | /n*pi*x\ | (8 - 4*x)*cos|------| dx | \ 2 / | / 1
$$\int\limits_{1}^{2} \left(8 - 4 x\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx$$
Integral((8 - 4*x)*cos(((n*pi)*x)/2), (x, 1, 2))
Respuesta (Indefinida)
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// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ || | // x for n = 0\ // x for n = 0\
| || // /pi*n*x\ \ | || | || |
| /n*pi*x\ || ||-2*cos|------| | | || /pi*n*x\ | || /pi*n*x\ |
| (8 - 4*x)*cos|------| dx = C + 4*|< || \ 2 / pi*n | | + 8*|<2*sin|------| | - 4*x*|<2*sin|------| |
| \ 2 / ||2*|<-------------- for ---- != 0| | || \ 2 / | || \ 2 / |
| || || pi*n 2 | | ||------------- otherwise| ||------------- otherwise|
/ || || | | \\ pi*n / \\ pi*n /
|| \\ 0 otherwise / |
||---------------------------------- otherwise|
|| pi*n |
\\ /
$$\int \left(8 - 4 x\right) \cos{\left(\frac{x \pi n}{2} \right)}\, dx = C - 4 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + 8 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{2 \sin{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{otherwise} \end{cases}\right) + 4 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{2 \left(\begin{cases} - \frac{2 \cos{\left(\frac{\pi n x}{2} \right)}}{\pi n} & \text{for}\: \frac{\pi n}{2} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{\pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ /pi*n\ /pi*n\ | 8*sin|----| 16*cos|----| | 16*cos(pi*n) \ 2 / \ 2 / |- ------------ - ----------- + ------------ for And(n > -oo, n < oo, n != 0) < 2 2 pi*n 2 2 | pi *n pi *n | | 2 otherwise \
$$\begin{cases} - \frac{8 \sin{\left(\frac{\pi n}{2} \right)}}{\pi n} + \frac{16 \cos{\left(\frac{\pi n}{2} \right)}}{\pi^{2} n^{2}} - \frac{16 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\2 & \text{otherwise} \end{cases}$$
=
=
/ /pi*n\ /pi*n\ | 8*sin|----| 16*cos|----| | 16*cos(pi*n) \ 2 / \ 2 / |- ------------ - ----------- + ------------ for And(n > -oo, n < oo, n != 0) < 2 2 pi*n 2 2 | pi *n pi *n | | 2 otherwise \
$$\begin{cases} - \frac{8 \sin{\left(\frac{\pi n}{2} \right)}}{\pi n} + \frac{16 \cos{\left(\frac{\pi n}{2} \right)}}{\pi^{2} n^{2}} - \frac{16 \cos{\left(\pi n \right)}}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\2 & \text{otherwise} \end{cases}$$
Piecewise((-16*cos(pi*n)/(pi^2*n^2) - 8*sin(pi*n/2)/(pi*n) + 16*cos(pi*n/2)/(pi^2*n^2), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (2, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.