Sr Examen
Integral de x^2/(5-x^2) dx
Solución
Ha introducido
[src]
1 / | | 2 | x | ------ dx | 2 | 5 - x | / 0
$$\int\limits_{0}^{1} \frac{x^{2}}{5 - x^{2}}\, dx$$
Integral(x^2/(5 - x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___\ \
|| ___ |x*\/ 5 | |
/ ||-\/ 5 *acoth|-------| |
| || \ 5 / 2 |
| 2 ||---------------------- for x > 5|
| x || 5 |
| ------ dx = C - x - 5*|< |
| 2 || / ___\ |
| 5 - x || ___ |x*\/ 5 | |
| ||-\/ 5 *atanh|-------| |
/ || \ 5 / 2 |
||---------------------- for x < 5|
\\ 5 /
$$\int \frac{x^{2}}{5 - x^{2}}\, dx = C - x - 5 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} > 5 \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{\sqrt{5} x}{5} \right)}}{5} & \text{for}\: x^{2} < 5 \end{cases}\right)$$
Gráfica
Respuesta
[src]
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 5 *\pi*I + log\\/ 5 // \/ 5 *log\1 + \/ 5 / \/ 5 *\pi*I + log\-1 + \/ 5 // \/ 5 *log\\/ 5 /
-1 + ------------------------- + -------------------- - ------------------------------ - ----------------
2 2 2 2
$$-1 - \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{2} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{2} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{2} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{2}$$
=
=
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 5 *\pi*I + log\\/ 5 // \/ 5 *log\1 + \/ 5 / \/ 5 *\pi*I + log\-1 + \/ 5 // \/ 5 *log\\/ 5 /
-1 + ------------------------- + -------------------- - ------------------------------ - ----------------
2 2 2 2
$$-1 - \frac{\sqrt{5} \log{\left(\sqrt{5} \right)}}{2} + \frac{\sqrt{5} \log{\left(1 + \sqrt{5} \right)}}{2} - \frac{\sqrt{5} \left(\log{\left(-1 + \sqrt{5} \right)} + i \pi\right)}{2} + \frac{\sqrt{5} \left(\log{\left(\sqrt{5} \right)} + i \pi\right)}{2}$$
-1 + sqrt(5)*(pi*i + log(sqrt(5)))/2 + sqrt(5)*log(1 + sqrt(5))/2 - sqrt(5)*(pi*i + log(-1 + sqrt(5)))/2 - sqrt(5)*log(sqrt(5))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.