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