Sr Examen
Integral de cos(p^2*x)/x^(2/3) dx
Solución
Ha introducido
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oo / | | / 2 \ | cos\p *x/ | --------- dx | 2/3 | x | / 1
$$\int\limits_{1}^{\infty} \frac{\cos{\left(p^{2} x \right)}}{x^{\frac{2}{3}}}\, dx$$
Integral(cos(p^2*x)/x^(2/3), (x, 1, oo))
Respuesta (Indefinida)
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/ / | | | / 2 \ | / 2 \ | cos\p *x/ | cos\p *x/ | --------- dx = C + | --------- dx | 2/3 | 2/3 | x | x | | / /
$$\int \frac{\cos{\left(p^{2} x \right)}}{x^{\frac{2}{3}}}\, dx = C + \int \frac{\cos{\left(p^{2} x \right)}}{x^{\frac{2}{3}}}\, dx$$
Respuesta
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/ / \ | | _ / | 4 \| | | |_ | 1/6 | -p || | |3 ___ Gamma(-1/6)* | | | ----|| | ____ |\/ 2 *Gamma(1/6) 1 2 \1/2, 7/6 | 4 /| |\/ pi *|---------------- + ----------------------------------| | | 2/3 ____ | | \p *Gamma(1/3) \/ pi *Gamma(5/6) / |-------------------------------------------------------------- for 4*|arg(p)| = 0 | 2 | < oo | / | | | | / 2\ | | cos\x*p / | | --------- dx otherwise | | 2/3 | | x | | | / | 1 \
$$\begin{cases} \frac{\sqrt{\pi} \left(\frac{\Gamma\left(- \frac{1}{6}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{6} \\ \frac{1}{2}, \frac{7}{6} \end{matrix}\middle| {- \frac{p^{4}}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{5}{6}\right)} + \frac{\sqrt[3]{2} \Gamma\left(\frac{1}{6}\right)}{p^{\frac{2}{3}} \Gamma\left(\frac{1}{3}\right)}\right)}{2} & \text{for}\: 4 \left|{\arg{\left(p \right)}}\right| = 0 \\\int\limits_{1}^{\infty} \frac{\cos{\left(p^{2} x \right)}}{x^{\frac{2}{3}}}\, dx & \text{otherwise} \end{cases}$$
=
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/ / \ | | _ / | 4 \| | | |_ | 1/6 | -p || | |3 ___ Gamma(-1/6)* | | | ----|| | ____ |\/ 2 *Gamma(1/6) 1 2 \1/2, 7/6 | 4 /| |\/ pi *|---------------- + ----------------------------------| | | 2/3 ____ | | \p *Gamma(1/3) \/ pi *Gamma(5/6) / |-------------------------------------------------------------- for 4*|arg(p)| = 0 | 2 | < oo | / | | | | / 2\ | | cos\x*p / | | --------- dx otherwise | | 2/3 | | x | | | / | 1 \
$$\begin{cases} \frac{\sqrt{\pi} \left(\frac{\Gamma\left(- \frac{1}{6}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{1}{6} \\ \frac{1}{2}, \frac{7}{6} \end{matrix}\middle| {- \frac{p^{4}}{4}} \right)}}{\sqrt{\pi} \Gamma\left(\frac{5}{6}\right)} + \frac{\sqrt[3]{2} \Gamma\left(\frac{1}{6}\right)}{p^{\frac{2}{3}} \Gamma\left(\frac{1}{3}\right)}\right)}{2} & \text{for}\: 4 \left|{\arg{\left(p \right)}}\right| = 0 \\\int\limits_{1}^{\infty} \frac{\cos{\left(p^{2} x \right)}}{x^{\frac{2}{3}}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((sqrt(pi)*(2^(1/3)*gamma(1/6)/(p^(2/3)*gamma(1/3)) + gamma(-1/6)*hyper((1/6,), (1/2, 7/6), -p^4/4)/(sqrt(pi)*gamma(5/6)))/2, 4*Abs(arg(p)) = 0), (Integral(cos(x*p^2)/x^(2/3), (x, 1, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.