Sr Examen
Integral de x^n*(e^((-2)/x^2)-cos(2/x)) dx
Solución
Ha introducido
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oo / | | / -2 \ | | --- | | | 2 | | n | x /2\| | x *|E - cos|-|| dx | \ \x// | / 1
$$\int\limits_{1}^{\infty} x^{n} \left(- \cos{\left(\frac{2}{x} \right)} + e^{- \frac{2}{x^{2}}}\right)\, dx$$
Integral(x^n*(E^(-2/x^2) - cos(2/x)), (x, 1, oo))
Respuesta (Indefinida)
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/ 1 n | \
/ _ | - - - - | -1 | n n
| n / 1 n\ |_ | 2 2 | ---| - -
| / -2 \ x*x *Gamma|- - - -|* | | | 2| ___ 2 / 1 n\ / 1 n 2 \ ___ 2 / 1 n\ / 1 n 2 \
| | --- | \ 2 2/ 1 2 | 1 n | x | \/ 2 *2 *Gamma|- - - -|*lowergamma|- - - -, --| n*\/ 2 *2 *Gamma|- - - -|*lowergamma|- - - -, --|
| | 2 | |1/2, - - - | | \ 2 2/ | 2 2 2| \ 2 2/ | 2 2 2|
| n | x /2\| \ 2 2 | / \ x / \ x /
| x *|E - cos|-|| dx = C + ------------------------------------------- + ----------------------------------------------- + -------------------------------------------------
| \ \x// /1 n\ /1 n\ /1 n\
| 2*Gamma|- - -| 4*Gamma|- - -| 4*Gamma|- - -|
/ \2 2/ \2 2/ \2 2/
$$\int x^{n} \left(- \cos{\left(\frac{2}{x} \right)} + e^{- \frac{2}{x^{2}}}\right)\, dx = \frac{\sqrt{2} \cdot 2^{\frac{n}{2}} n \Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) \gamma\left(- \frac{n}{2} - \frac{1}{2}, \frac{2}{x^{2}}\right)}{4 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} + \frac{\sqrt{2} \cdot 2^{\frac{n}{2}} \Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) \gamma\left(- \frac{n}{2} - \frac{1}{2}, \frac{2}{x^{2}}\right)}{4 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} + C + \frac{x x^{n} \Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} - \frac{n}{2} - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - \frac{n}{2} \end{matrix}\middle| {- \frac{1}{x^{2}}} \right)}}{2 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)}$$
Respuesta
[src]
/ | / 1 n | \ | _ | - - - - | | | / 1 n\ |_ | 2 2 | | 1 n /1 n\ / 1 n\ | Gamma|- - - -|* | | | -1| - + - 2*pi*I*|- + -| + 2*pi*I*|- - - -| | \ 2 2/ 1 2 | 1 n | | 2 2 / 1 n\ \2 2/ \ 2 2/ / 1 n\ / 1 n \ | |1/2, - - - | | 2 *|- - - -|*e *Gamma|- - - -|*lowergamma|- - - -, 2| | \ 2 2 | / \ 2 2/ \ 2 2/ \ 2 2 / / re(n) 5 re(n) \ |- ------------------------------------- + ----------------------------------------------------------------------------------------- for And|2 + ----- < 2, - + ----- < 2| | /1 n\ /1 n\ \ 2 2 2 / | 2*Gamma|- - -| 2*Gamma|- - -| | \2 2/ \2 2/ < | oo | / | | | | / -2 \ | | | ---| | | | 2| | | n | /2\ x | | | x *|- cos|-| + e | dx otherwise | | \ \x/ / | | | / \ 1
$$\begin{cases} \frac{2^{\frac{n}{2} + \frac{1}{2}} \left(- \frac{n}{2} - \frac{1}{2}\right) e^{2 i \pi \left(- \frac{n}{2} - \frac{1}{2}\right) + 2 i \pi \left(\frac{n}{2} + \frac{1}{2}\right)} \Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) \gamma\left(- \frac{n}{2} - \frac{1}{2}, 2\right)}{2 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} - \frac{\Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} - \frac{n}{2} - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - \frac{n}{2} \end{matrix}\middle| {-1} \right)}}{2 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(n\right)}}{2} + 2 < 2 \wedge \frac{\operatorname{re}{\left(n\right)}}{2} + \frac{5}{2} < 2 \\\int\limits_{1}^{\infty} x^{n} \left(- \cos{\left(\frac{2}{x} \right)} + e^{- \frac{2}{x^{2}}}\right)\, dx & \text{otherwise} \end{cases}$$
=
=
/ | / 1 n | \ | _ | - - - - | | | / 1 n\ |_ | 2 2 | | 1 n /1 n\ / 1 n\ | Gamma|- - - -|* | | | -1| - + - 2*pi*I*|- + -| + 2*pi*I*|- - - -| | \ 2 2/ 1 2 | 1 n | | 2 2 / 1 n\ \2 2/ \ 2 2/ / 1 n\ / 1 n \ | |1/2, - - - | | 2 *|- - - -|*e *Gamma|- - - -|*lowergamma|- - - -, 2| | \ 2 2 | / \ 2 2/ \ 2 2/ \ 2 2 / / re(n) 5 re(n) \ |- ------------------------------------- + ----------------------------------------------------------------------------------------- for And|2 + ----- < 2, - + ----- < 2| | /1 n\ /1 n\ \ 2 2 2 / | 2*Gamma|- - -| 2*Gamma|- - -| | \2 2/ \2 2/ < | oo | / | | | | / -2 \ | | | ---| | | | 2| | | n | /2\ x | | | x *|- cos|-| + e | dx otherwise | | \ \x/ / | | | / \ 1
$$\begin{cases} \frac{2^{\frac{n}{2} + \frac{1}{2}} \left(- \frac{n}{2} - \frac{1}{2}\right) e^{2 i \pi \left(- \frac{n}{2} - \frac{1}{2}\right) + 2 i \pi \left(\frac{n}{2} + \frac{1}{2}\right)} \Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) \gamma\left(- \frac{n}{2} - \frac{1}{2}, 2\right)}{2 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} - \frac{\Gamma\left(- \frac{n}{2} - \frac{1}{2}\right) {{}_{1}F_{2}\left(\begin{matrix} - \frac{n}{2} - \frac{1}{2} \\ \frac{1}{2}, \frac{1}{2} - \frac{n}{2} \end{matrix}\middle| {-1} \right)}}{2 \Gamma\left(\frac{1}{2} - \frac{n}{2}\right)} & \text{for}\: \frac{\operatorname{re}{\left(n\right)}}{2} + 2 < 2 \wedge \frac{\operatorname{re}{\left(n\right)}}{2} + \frac{5}{2} < 2 \\\int\limits_{1}^{\infty} x^{n} \left(- \cos{\left(\frac{2}{x} \right)} + e^{- \frac{2}{x^{2}}}\right)\, dx & \text{otherwise} \end{cases}$$
Piecewise((-gamma(-1/2 - n/2)*hyper((-1/2 - n/2,), (1/2, 1/2 - n/2), -1)/(2*gamma(1/2 - n/2)) + 2^(1/2 + n/2)*(-1/2 - n/2)*exp(2*pi*i*(1/2 + n/2) + 2*pi*i*(-1/2 - n/2))*gamma(-1/2 - n/2)*lowergamma(-1/2 - n/2, 2)/(2*gamma(1/2 - n/2)), (2 + re(n)/2 < 2)∧(5/2 + re(n)/2 < 2)), (Integral(x^n*(-cos(2/x) + exp(-2/x^2)), (x, 1, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.