Sr Examen
Integral de x^2/(2-x^2)^2 dx
Solución
Ha introducido
[src]
1 / | | 2 | x | --------- dx | 2 | / 2\ | \2 - x / | / 0
$$\int\limits_{0}^{1} \frac{x^{2}}{\left(2 - x^{2}\right)^{2}}\, dx$$
Integral(x^2/(2 - x^2)^2, (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___\ \
// / / ___\ / ___\\ \ || ___ |x*\/ 2 | |
/ || | | x*\/ 2 | | x*\/ 2 || | ||-\/ 2 *acoth|-------| |
| || | log|-1 + -------| log|1 + -------|| | || \ 2 / 2 |
| 2 || ___ | 1 1 \ 2 / \ 2 /| | ||---------------------- for x > 2|
| x ||\/ 2 *|- --------------- - ---------------- - ----------------- + ----------------| | || 2 |
| --------- dx = C + 2*|< | / ___\ / ___\ 4 4 | | + |< |
| 2 || | | x*\/ 2 | | x*\/ 2 | | | || / ___\ |
| / 2\ || | 4*|1 + -------| 4*|-1 + -------| | | || ___ |x*\/ 2 | |
| \2 - x / || \ \ 2 / \ 2 / / / ___ ___\| ||-\/ 2 *atanh|-------| |
| ||----------------------------------------------------------------------------------- for And\x > -\/ 2 , x < \/ 2 /| || \ 2 / 2 |
/ \\ 4 / ||---------------------- for x < 2|
\\ 2 /
$$\int \frac{x^{2}}{\left(2 - x^{2}\right)^{2}}\, dx = C + 2 \left(\begin{cases} \frac{\sqrt{2} \left(- \frac{\log{\left(\frac{\sqrt{2} x}{2} - 1 \right)}}{4} + \frac{\log{\left(\frac{\sqrt{2} x}{2} + 1 \right)}}{4} - \frac{1}{4 \left(\frac{\sqrt{2} x}{2} + 1\right)} - \frac{1}{4 \left(\frac{\sqrt{2} x}{2} - 1\right)}\right)}{4} & \text{for}\: x > - \sqrt{2} \wedge x < \sqrt{2} \end{cases}\right) + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} & \text{for}\: x^{2} < 2 \end{cases}$$
Gráfica
Respuesta
[src]
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\ 1 \/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 / - - ------------------------- - -------------------- + ------------------------------ + ---------------- 2 8 8 8 8
$$- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{8} + \frac{1}{2} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{8}$$
=
=
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\ 1 \/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 / - - ------------------------- - -------------------- + ------------------------------ + ---------------- 2 8 8 8 8
$$- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{8} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{8} + \frac{1}{2} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{8} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{8}$$
1/2 - sqrt(2)*(pi*i + log(sqrt(2)))/8 - sqrt(2)*log(1 + sqrt(2))/8 + sqrt(2)*(pi*i + log(-1 + sqrt(2)))/8 + sqrt(2)*log(sqrt(2))/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.