Sr Examen
Integral de -4*(x^2)*cos(n*x) dx
Solución
Ha introducido
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-1 / | | 2 | -4*x *cos(n*x) dx | / -pi
$$\int\limits_{- \pi}^{-1} - 4 x^{2} \cos{\left(n x \right)}\, dx$$
Integral((-4*x^2)*cos(n*x), (x, -pi, -1))
Respuesta (Indefinida)
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// 3 \
|| x |
|| -- for n = 0|
|| 3 |
/ || |
| ||/sin(n*x) x*cos(n*x) | // x for n = 0\
| 2 |||-------- - ---------- for n != 0 | 2 || |
| -4*x *cos(n*x) dx = C + 8*|<| 2 n | - 4*x *|
$$\int - 4 x^{2} \cos{\left(n x \right)}\, dx = C - 4 x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + 8 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(n x \right)}}{n} + \frac{\sin{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
/ 2 | 8*sin(n) 4*sin(n) 8*sin(pi*n) 8*cos(n) 8*pi*cos(pi*n) 4*pi *sin(pi*n) |- -------- + -------- + ----------- + -------- - -------------- - --------------- for And(n > -oo, n < oo, n != 0) | 3 n 3 2 2 n | n n n n < | 3 | 4 4*pi | - - ----- otherwise | 3 3 \
$$\begin{cases} \frac{4 \sin{\left(n \right)}}{n} - \frac{4 \pi^{2} \sin{\left(\pi n \right)}}{n} + \frac{8 \cos{\left(n \right)}}{n^{2}} - \frac{8 \pi \cos{\left(\pi n \right)}}{n^{2}} - \frac{8 \sin{\left(n \right)}}{n^{3}} + \frac{8 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} - \frac{4 \pi^{3}}{3} & \text{otherwise} \end{cases}$$
=
=
/ 2 | 8*sin(n) 4*sin(n) 8*sin(pi*n) 8*cos(n) 8*pi*cos(pi*n) 4*pi *sin(pi*n) |- -------- + -------- + ----------- + -------- - -------------- - --------------- for And(n > -oo, n < oo, n != 0) | 3 n 3 2 2 n | n n n n < | 3 | 4 4*pi | - - ----- otherwise | 3 3 \
$$\begin{cases} \frac{4 \sin{\left(n \right)}}{n} - \frac{4 \pi^{2} \sin{\left(\pi n \right)}}{n} + \frac{8 \cos{\left(n \right)}}{n^{2}} - \frac{8 \pi \cos{\left(\pi n \right)}}{n^{2}} - \frac{8 \sin{\left(n \right)}}{n^{3}} + \frac{8 \sin{\left(\pi n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{4}{3} - \frac{4 \pi^{3}}{3} & \text{otherwise} \end{cases}$$
Piecewise((-8*sin(n)/n^3 + 4*sin(n)/n + 8*sin(pi*n)/n^3 + 8*cos(n)/n^2 - 8*pi*cos(pi*n)/n^2 - 4*pi^2*sin(pi*n)/n, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4/3 - 4*pi^3/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.