Sr Examen
Integral de xroot|1+2/x| dx
Solución
Ha introducido
[src]
1 / | | _________ | / | 2| | x* / |1 + -| dx | \/ | x| | / 0
$$\int\limits_{0}^{1} x \sqrt{\left|{1 + \frac{2}{x}}\right|}\, dx$$
Integral(x*sqrt(|1 + 2/x|), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / _______\ _______ _______ \
|| ___ | / 2 | ___ /1\ ___ / 2 ___ 2 / 2 |
|| ___ pi*\/ 2 *log|1 + / 1 + - | ___ 2 ___ 2 pi*\/ 2 *log|-| ___ pi*x*\/ 2 * / 1 + - pi*\/ 2 *x * / 1 + - |
|| pi*x*\/ 2 \ \/ x / pi*\/ 2 *x pi*\/ 2 *x \x/ pi*x*\/ 2 \/ x \/ x 2 |
||----------------------- + ----------------------------- + ------------------------- + ------------------------- + ----------------------- - ---------------------- - ------------------------- - ------------------------- for --- < 1|
/ ||Gamma(-9/4)*Gamma(13/4) Gamma(-9/4)*Gamma(13/4) 2*Gamma(-9/4)*Gamma(13/4) 2*Gamma(-7/4)*Gamma(11/4) 2*Gamma(1/4)*Gamma(3/4) Gamma(-3/4)*Gamma(7/4) 2*Gamma(-9/4)*Gamma(13/4) 2*Gamma(-9/4)*Gamma(13/4) |x| |
| || |
| _________ || / ___ ___\ |
| / | 2| || ___ |\/ 2 *\/ x | |
| x* / |1 + -| dx = C + |< pi*I*\/ 2 *asinh|-----------| ___ ___ ___ 3/2 ___ 5/2 |
| \/ | x| || \ 2 / pi*I*\/ 2 *\/ x 3*pi*I*\/ 2 *x pi*I*\/ 2 *x |x| |
| || ----------------------------- - ------------------------------- - --------------------------------- - --------------------------------- for --- < 1|
/ || Gamma(1/4)*Gamma(3/4) _______ _______ _______ 2 |
|| \/ 2 + x *Gamma(1/4)*Gamma(3/4) 2*\/ 2 + x *Gamma(1/4)*Gamma(3/4) 2*\/ 2 + x *Gamma(1/4)*Gamma(3/4) |
|| |
|| / | -pi*I\ |
|| ___ ____ __2, 1 |-1/2 -5/4, 1 | 2*e | |
|| -2*\/ 2 *\/ pi */__ | | --------| otherwise |
\\ \_|3, 3 \-2, 0 -5/4 | x / /
$$\int x \sqrt{\left|{1 + \frac{2}{x}}\right|}\, dx = C + \begin{cases} - \frac{\sqrt{2} \pi x^{2} \sqrt{1 + \frac{2}{x}}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x^{2}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x^{2}}{2 \Gamma\left(- \frac{7}{4}\right) \Gamma\left(\frac{11}{4}\right)} - \frac{\sqrt{2} \pi x \sqrt{1 + \frac{2}{x}}}{2 \Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} + \frac{\sqrt{2} \pi x}{\Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} - \frac{\sqrt{2} \pi x}{\Gamma\left(- \frac{3}{4}\right) \Gamma\left(\frac{7}{4}\right)} + \frac{\sqrt{2} \pi \log{\left(\frac{1}{x} \right)}}{2 \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} + \frac{\sqrt{2} \pi \log{\left(\sqrt{1 + \frac{2}{x}} + 1 \right)}}{\Gamma\left(- \frac{9}{4}\right) \Gamma\left(\frac{13}{4}\right)} & \text{for}\: \frac{2}{\left|{x}\right|} < 1 \\- \frac{\sqrt{2} i \pi x^{\frac{5}{2}}}{2 \sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} - \frac{3 \sqrt{2} i \pi x^{\frac{3}{2}}}{2 \sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} - \frac{\sqrt{2} i \pi \sqrt{x}}{\sqrt{x + 2} \Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} + \frac{\sqrt{2} i \pi \operatorname{asinh}{\left(\frac{\sqrt{2} \sqrt{x}}{2} \right)}}{\Gamma\left(\frac{1}{4}\right) \Gamma\left(\frac{3}{4}\right)} & \text{for}\: \frac{\left|{x}\right|}{2} < 1 \\- 2 \sqrt{2} \sqrt{\pi} {G_{3, 3}^{2, 1}\left(\begin{matrix} - \frac{1}{2} & - \frac{5}{4}, 1 \\-2, 0 & - \frac{5}{4} \end{matrix} \middle| {\frac{2 e^{- i \pi}}{x}} \right)} & \text{otherwise} \end{cases}$$
Gráfica
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.