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