Sr Examen
Integral de x^(2*a)*sin(1/x^2) dx
Solución
Ha introducido
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oo / | | 2*a /1 \ | x *sin|--| dx | | 2| | \x / | / 1
$$\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx$$
Integral(x^(2*a)*sin(1/(x^2)), (x, 1, oo))
Respuesta (Indefinida)
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/ 1 a | \
_ | - - - | -1 |
2*a /1 a\ |_ | 4 2 | ----|
x *Gamma|- - -|* | | | 4|
/ \4 2/ 1 2 | 5 a | 4*x |
| |3/2, - - - | |
| 2*a /1 \ \ 4 2 | /
| x *sin|--| dx = C - ------------------------------------------
| | 2| /5 a\
| \x / 4*x*Gamma|- - -|
| \4 2/
/
$$\int x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx = C - \frac{x^{2 a} \Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4 x^{4}}} \right)}}{4 x \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)}$$
Respuesta
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/ | / 1 a | \ | _ | - - - | | | /1 a\ |_ | 4 2 | | |Gamma|- - -|* | | | -1/4| | \4 2/ 1 2 | 5 a | | | |3/2, - - - | | | \ 4 2 | / / re(a) 5 re(a) 3 re(a) 7 re(a) \ |------------------------------------- for And|1 + ----- < 2, - + ----- < 2, - + ----- < 2, - + ----- < 2| | /5 a\ \ 2 4 2 2 2 4 2 / | 4*Gamma|- - -| < \4 2/ | | oo | / | | | | 2*a /1 \ | | x *sin|--| dx otherwise | | | 2| | | \x / | | | / \ 1
$$\begin{cases} \frac{\Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{4 \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(a\right)}}{2} + 1 < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{5}{4} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{3}{2} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{7}{4} < 2 \\\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx & \text{otherwise} \end{cases}$$
=
=
/ | / 1 a | \ | _ | - - - | | | /1 a\ |_ | 4 2 | | |Gamma|- - -|* | | | -1/4| | \4 2/ 1 2 | 5 a | | | |3/2, - - - | | | \ 4 2 | / / re(a) 5 re(a) 3 re(a) 7 re(a) \ |------------------------------------- for And|1 + ----- < 2, - + ----- < 2, - + ----- < 2, - + ----- < 2| | /5 a\ \ 2 4 2 2 2 4 2 / | 4*Gamma|- - -| < \4 2/ | | oo | / | | | | 2*a /1 \ | | x *sin|--| dx otherwise | | | 2| | | \x / | | | / \ 1
$$\begin{cases} \frac{\Gamma\left(\frac{1}{4} - \frac{a}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{4} - \frac{a}{2} \\ \frac{3}{2}, \frac{5}{4} - \frac{a}{2} \end{matrix}\middle| {- \frac{1}{4}} \right)}}{4 \Gamma\left(\frac{5}{4} - \frac{a}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(a\right)}}{2} + 1 < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{5}{4} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{3}{2} < 2 \wedge \frac{\operatorname{re}{\left(a\right)}}{2} + \frac{7}{4} < 2 \\\int\limits_{1}^{\infty} x^{2 a} \sin{\left(\frac{1}{x^{2}} \right)}\, dx & \text{otherwise} \end{cases}$$
Piecewise((gamma(1/4 - a/2)*hyper((1/4 - a/2,), (3/2, 5/4 - a/2), -1/4)/(4*gamma(5/4 - a/2)), (1 + re(a)/2 < 2)∧(5/4 + re(a)/2 < 2)∧(3/2 + re(a)/2 < 2)∧(7/4 + re(a)/2 < 2)), (Integral(x^(2*a)*sin(x^(-2)), (x, 1, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.