Sr Examen
Integral de (2x+3)*cos((2pi*x)/3n) dx
Solución
Ha introducido
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1 / | | /2*pi*x \ | (2*x + 3)*cos|------*n| dx | \ 3 / | / 0
$$\int\limits_{0}^{1} \left(2 x + 3\right) \cos{\left(n \frac{2 \pi x}{3} \right)}\, dx$$
Integral((2*x + 3)*cos((((2*pi)*x)/3)*n), (x, 0, 1))
Respuesta (Indefinida)
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// 2 \
|| x |
|| -- for n = 0|
|| 2 |
/ || | // x for n = 0\ // x for n = 0\
| || // /2*pi*n*x\ \ | || | || |
| /2*pi*x \ || ||-3*cos|--------| | | || /2*pi*n*x\ | || /2*pi*n*x\ |
| (2*x + 3)*cos|------*n| dx = C - 2*|< || \ 3 / 2*pi*n | | + 3*|<3*sin|--------| | + 2*x*|<3*sin|--------| |
| \ 3 / ||3*|<---------------- for ------ != 0| | || \ 3 / | || \ 3 / |
| || || 2*pi*n 3 | | ||--------------- otherwise| ||--------------- otherwise|
/ || || | | \\ 2*pi*n / \\ 2*pi*n /
|| \\ 0 otherwise / |
||-------------------------------------- otherwise|
|| 2*pi*n |
\\ /
$$\int \left(2 x + 3\right) \cos{\left(n \frac{2 \pi x}{3} \right)}\, dx = C + 2 x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) + 3 \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{3 \sin{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{3 \left(\begin{cases} - \frac{3 \cos{\left(\frac{2 \pi n x}{3} \right)}}{2 \pi n} & \text{for}\: \frac{2 \pi n}{3} \neq 0 \\0 & \text{otherwise} \end{cases}\right)}{2 \pi n} & \text{otherwise} \end{cases}\right)$$
Respuesta
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/ /2*pi*n\ /2*pi*n\ | 9*cos|------| 15*sin|------| | 9 \ 3 / \ 3 / |- -------- + ------------- + -------------- for And(n > -oo, n < oo, n != 0) < 2 2 2 2 2*pi*n | 2*pi *n 2*pi *n | | 4 otherwise \
$$\begin{cases} \frac{15 \sin{\left(\frac{2 \pi n}{3} \right)}}{2 \pi n} + \frac{9 \cos{\left(\frac{2 \pi n}{3} \right)}}{2 \pi^{2} n^{2}} - \frac{9}{2 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases}$$
=
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/ /2*pi*n\ /2*pi*n\ | 9*cos|------| 15*sin|------| | 9 \ 3 / \ 3 / |- -------- + ------------- + -------------- for And(n > -oo, n < oo, n != 0) < 2 2 2 2 2*pi*n | 2*pi *n 2*pi *n | | 4 otherwise \
$$\begin{cases} \frac{15 \sin{\left(\frac{2 \pi n}{3} \right)}}{2 \pi n} + \frac{9 \cos{\left(\frac{2 \pi n}{3} \right)}}{2 \pi^{2} n^{2}} - \frac{9}{2 \pi^{2} n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\4 & \text{otherwise} \end{cases}$$
Piecewise((-9/(2*pi^2*n^2) + 9*cos(2*pi*n/3)/(2*pi^2*n^2) + 15*sin(2*pi*n/3)/(2*pi*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (4, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.