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