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