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