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