Sr Examen
Integral de (x^2-x)cos(n*x) dx
Solución
Ha introducido
[src]
pi / | | / 2 \ | \x - x/*cos(n*x) dx | / 0
$$\int\limits_{0}^{\pi} \left(x^{2} - x\right) \cos{\left(n x \right)}\, dx$$
Integral((x^2 - x)*cos(n*x), (x, 0, pi))
Respuesta (Indefinida)
[src]
// 3 \
|| x | // 2 \
|| -- for n = 0| || x |
|| 3 | || -- for n = 0|
/ || | || 2 |
| ||/sin(n*x) x*cos(n*x) | // x for n = 0\ // x for n = 0\ || |
| / 2 \ |||-------- - ---------- for n != 0 | 2 || | || | ||/-cos(n*x) |
| \x - x/*cos(n*x) dx = C - 2*|<| 2 n | + x *|
$$\int \left(x^{2} - x\right) \cos{\left(n x \right)}\, dx = C + x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + \begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases} - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(n x \right)}}{n} + \frac{\sin{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ 2 |1 cos(pi*n) 2*sin(pi*n) pi *sin(pi*n) pi*sin(pi*n) 2*pi*cos(pi*n) |-- - --------- - ----------- + ------------- - ------------ + -------------- for And(n > -oo, n < oo, n != 0) | 2 2 3 n n 2 |n n n n < | 2 3 | pi pi | - --- + --- otherwise | 2 3 \
$$\begin{cases} - \frac{\pi \sin{\left(\pi n \right)}}{n} + \frac{\pi^{2} \sin{\left(\pi n \right)}}{n} - \frac{\cos{\left(\pi n \right)}}{n^{2}} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{2}} + \frac{1}{n^{2}} - \frac{2 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{\pi^{2}}{2} + \frac{\pi^{3}}{3} & \text{otherwise} \end{cases}$$
=
=
/ 2 |1 cos(pi*n) 2*sin(pi*n) pi *sin(pi*n) pi*sin(pi*n) 2*pi*cos(pi*n) |-- - --------- - ----------- + ------------- - ------------ + -------------- for And(n > -oo, n < oo, n != 0) | 2 2 3 n n 2 |n n n n < | 2 3 | pi pi | - --- + --- otherwise | 2 3 \
$$\begin{cases} - \frac{\pi \sin{\left(\pi n \right)}}{n} + \frac{\pi^{2} \sin{\left(\pi n \right)}}{n} - \frac{\cos{\left(\pi n \right)}}{n^{2}} + \frac{2 \pi \cos{\left(\pi n \right)}}{n^{2}} + \frac{1}{n^{2}} - \frac{2 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{\pi^{2}}{2} + \frac{\pi^{3}}{3} & \text{otherwise} \end{cases}$$
Piecewise((n^(-2) - cos(pi*n)/n^2 - 2*sin(pi*n)/n^3 + pi^2*sin(pi*n)/n - pi*sin(pi*n)/n + 2*pi*cos(pi*n)/n^2, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-pi^2/2 + pi^3/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.