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