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