Sr Examen
Integral de (x-3)*cos(pi*n*x) dx
Solución
Ha introducido
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1 / | | (x - 3)*cos(pi*n*x) dx | / 0
$$\int\limits_{0}^{1} \left(x - 3\right) \cos{\left(x \pi n \right)}\, dx$$
Integral((x - 3)*cos((pi*n)*x), (x, 0, 1))
Respuesta (Indefinida)
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// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ || | // x for n = 0\ // x for n = 0\
| ||/-cos(pi*n*x) | || | || |
| (x - 3)*cos(pi*n*x) dx = C - |<|------------- for pi*n != 0 | - 3*|
$$\int \left(x - 3\right) \cos{\left(x \pi n \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - 3 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(\pi n x \right)}}{\pi n} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(\pi n x \right)}}{\pi n} & \text{for}\: \pi n \neq 0 \\0 & \text{otherwise} \end{cases}}{\pi n} & \text{otherwise} \end{cases}$$
Respuesta
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/ 1 cos(pi*n) 2*sin(pi*n) |- ------ + --------- - ----------- for And(n > -oo, n < oo, n != 0) | 2 2 2 2 pi*n < pi *n pi *n | | -5/2 otherwise \
$$\begin{cases} - \frac{2 \sin{\left(\pi n \right)}}{\pi n} + \frac{\cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{1}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{5}{2} & \text{otherwise} \end{cases}$$
=
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/ 1 cos(pi*n) 2*sin(pi*n) |- ------ + --------- - ----------- for And(n > -oo, n < oo, n != 0) | 2 2 2 2 pi*n < pi *n pi *n | | -5/2 otherwise \
$$\begin{cases} - \frac{2 \sin{\left(\pi n \right)}}{\pi n} + \frac{\cos{\left(\pi n \right)}}{\pi^{2} n^{2}} - \frac{1}{\pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{5}{2} & \text{otherwise} \end{cases}$$
Piecewise((-1/(pi^2*n^2) + cos(pi*n)/(pi^2*n^2) - 2*sin(pi*n)/(pi*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-5/2, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.