Sr Examen
Integral de sinx/(x)^(a) dx
Solución
Ha introducido
[src]
oo / | | sin(x) | ------ dx | a | x | / 1
$$\int\limits_{1}^{\infty} \frac{\sin{\left(x \right)}}{x^{a}}\, dx$$
Integral(sin(x)/x^a, (x, 1, oo))
Respuesta (Indefinida)
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/ / | | | sin(x) | -a | ------ dx = C + | x *sin(x) dx | a | | x / | /
$$\int \frac{\sin{\left(x \right)}}{x^{a}}\, dx = C + \int x^{- a} \sin{\left(x \right)}\, dx$$
Respuesta
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/ / \ | | / a | \| | | _ | 1 - - | || | | / a\ |_ | 2 | || | | Gamma|-1 + -|* | | | -1/4|| | | 1 - a / a\ \ 2/ 1 2 | a | || | |2 *Gamma|1 - -| |3/2, 2 - - | || | ____ | \ 2/ \ 2 | /| |\/ pi *|------------------- + --------------------------------------| | | /1 a\ ____ /a\ | | | Gamma|- + -| \/ pi *Gamma|-| | < \ \2 2/ \2/ / |--------------------------------------------------------------------- for And(-3/2 + re(a) > -3/2, -1/2 + re(a) > -3/2) | 2 | | oo | / | | | | -a | | x *sin(x) dx otherwise | | | / \ 1
$$\begin{cases} \frac{\sqrt{\pi} \left(\frac{2^{1 - a} \Gamma\left(1 - \frac{a}{2}\right)}{\Gamma\left(\frac{a}{2} + \frac{1}{2}\right)} + \frac{\Gamma\left(\frac{a}{2} - 1\right) {{}_{1}F_{2}\left(\begin{matrix} 1 - \frac{a}{2} \\ \frac{3}{2}, 2 - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{a}{2}\right)}\right)}{2} & \text{for}\: \operatorname{re}{\left(a\right)} - \frac{3}{2} > - \frac{3}{2} \wedge \operatorname{re}{\left(a\right)} - \frac{1}{2} > - \frac{3}{2} \\\int\limits_{1}^{\infty} x^{- a} \sin{\left(x \right)}\, dx & \text{otherwise} \end{cases}$$
=
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/ / \ | | / a | \| | | _ | 1 - - | || | | / a\ |_ | 2 | || | | Gamma|-1 + -|* | | | -1/4|| | | 1 - a / a\ \ 2/ 1 2 | a | || | |2 *Gamma|1 - -| |3/2, 2 - - | || | ____ | \ 2/ \ 2 | /| |\/ pi *|------------------- + --------------------------------------| | | /1 a\ ____ /a\ | | | Gamma|- + -| \/ pi *Gamma|-| | < \ \2 2/ \2/ / |--------------------------------------------------------------------- for And(-3/2 + re(a) > -3/2, -1/2 + re(a) > -3/2) | 2 | | oo | / | | | | -a | | x *sin(x) dx otherwise | | | / \ 1
$$\begin{cases} \frac{\sqrt{\pi} \left(\frac{2^{1 - a} \Gamma\left(1 - \frac{a}{2}\right)}{\Gamma\left(\frac{a}{2} + \frac{1}{2}\right)} + \frac{\Gamma\left(\frac{a}{2} - 1\right) {{}_{1}F_{2}\left(\begin{matrix} 1 - \frac{a}{2} \\ \frac{3}{2}, 2 - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{a}{2}\right)}\right)}{2} & \text{for}\: \operatorname{re}{\left(a\right)} - \frac{3}{2} > - \frac{3}{2} \wedge \operatorname{re}{\left(a\right)} - \frac{1}{2} > - \frac{3}{2} \\\int\limits_{1}^{\infty} x^{- a} \sin{\left(x \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(pi)*(2^(1 - a)*gamma(1 - a/2)/gamma(1/2 + a/2) + gamma(-1 + a/2)*hyper((1 - a/2,), (3/2, 2 - a/2), -1/4)/(sqrt(pi)*gamma(a/2)))/2, (-3/2 + re(a) > -3/2)∧(-1/2 + re(a) > -3/2)), (Integral(x^(-a)*sin(x), (x, 1, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.