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