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