Sr Examen
Integral de xcos(2nx/a) dx
Solución
Ha introducido
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a / | | /2*n*x\ | x*cos|-----| dx | \ a / | / 0
$$\int\limits_{0}^{a} x \cos{\left(\frac{2 n x}{a} \right)}\, dx$$
Integral(x*cos(((2*n)*x)/a), (x, 0, a))
Respuesta (Indefinida)
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// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ || | // x for n = 0\
| || // /2*n*x\ \ | || |
| /2*n*x\ || ||-a*cos|-----| | | || /2*n*x\ |
| x*cos|-----| dx = C - |< || \ a / 2*n | | + x*|
$$\int x \cos{\left(\frac{2 n x}{a} \right)}\, dx = C + x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{a \sin{\left(\frac{2 n x}{a} \right)}}{2 n} & \text{otherwise} \end{cases}\right) - \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{a \left(\begin{cases} - \frac{a \cos{\left(\frac{2 n x}{a} \right)}}{2 n} & \text{for}\: \frac{2 n}{a} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{2 n} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/ 2 2 2 | a a *sin(2*n) a *cos(2*n) |- ---- + ----------- + ----------- for And(n > -oo, n < oo, n != 0) | 2 2*n 2 | 4*n 4*n < | 2 | a | -- otherwise | 2 \
$$\begin{cases} \frac{a^{2} \sin{\left(2 n \right)}}{2 n} + \frac{a^{2} \cos{\left(2 n \right)}}{4 n^{2}} - \frac{a^{2}}{4 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{a^{2}}{2} & \text{otherwise} \end{cases}$$
=
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/ 2 2 2 | a a *sin(2*n) a *cos(2*n) |- ---- + ----------- + ----------- for And(n > -oo, n < oo, n != 0) | 2 2*n 2 | 4*n 4*n < | 2 | a | -- otherwise | 2 \
$$\begin{cases} \frac{a^{2} \sin{\left(2 n \right)}}{2 n} + \frac{a^{2} \cos{\left(2 n \right)}}{4 n^{2}} - \frac{a^{2}}{4 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{a^{2}}{2} & \text{otherwise} \end{cases}$$
Piecewise((-a^2/(4*n^2) + a^2*sin(2*n)/(2*n) + a^2*cos(2*n)/(4*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.