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