Sr Examen
Integral de 1/(1+sqrt(x)-x) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | ___ | 1 + \/ x - x | / 0
$$\int\limits_{0}^{1} \frac{1}{- x + \left(\sqrt{x} + 1\right)}\, dx$$
Integral(1/(1 + sqrt(x) - x), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ / 1 ___\\ \
|| |2*\/ 5 *|- - + \/ x || |
|| ___ | \ 2 /| |
||-\/ 5 *acoth|---------------------| 2 |
/ || \ 5 / / 1 ___\ |
| ||------------------------------------ for |- - + \/ x | > 5/4|
| 1 / ___\ || 10 \ 2 / |
| ------------- dx = C - log\-1 + x - \/ x / - 4*|< |
| ___ || / ___ / 1 ___\\ |
| 1 + \/ x - x || |2*\/ 5 *|- - + \/ x || |
| || ___ | \ 2 /| |
/ ||-\/ 5 *atanh|---------------------| 2 |
|| \ 5 / / 1 ___\ |
||------------------------------------ for |- - + \/ x | < 5/4|
\\ 10 \ 2 / /
$$\int \frac{1}{- x + \left(\sqrt{x} + 1\right)}\, dx = C - 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(\sqrt{x} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\sqrt{x} - \frac{1}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(\sqrt{x} - \frac{1}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(\sqrt{x} - \frac{1}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) - \log{\left(- \sqrt{x} + x - 1 \right)}$$
Gráfica
Respuesta
[src]
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | 1 \/ 5 || ___ | 1 \/ 5 | ___ | |1 \/ 5 || ___ |1 \/ 5 |
\/ 5 *|pi*I + log|- - + -----|| \/ 5 *log|- - + -----| \/ 5 *|pi*I + log|- + -----|| \/ 5 *log|- + -----|
\ \ 2 2 // \ 2 2 / \ \2 2 // \2 2 /
- ------------------------------- - ---------------------- + ----------------------------- + --------------------
5 5 5 5
$$- \frac{\sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5} + \frac{\sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5}$$
=
=
/ / ___\\ / ___\ / / ___\\ / ___\
___ | | 1 \/ 5 || ___ | 1 \/ 5 | ___ | |1 \/ 5 || ___ |1 \/ 5 |
\/ 5 *|pi*I + log|- - + -----|| \/ 5 *log|- - + -----| \/ 5 *|pi*I + log|- + -----|| \/ 5 *log|- + -----|
\ \ 2 2 // \ 2 2 / \ \2 2 // \2 2 /
- ------------------------------- - ---------------------- + ----------------------------- + --------------------
5 5 5 5
$$- \frac{\sqrt{5} \log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} + \frac{\sqrt{5} \log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}}{5} - \frac{\sqrt{5} \left(\log{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5} + \frac{\sqrt{5} \left(\log{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)} + i \pi\right)}{5}$$
-sqrt(5)*(pi*i + log(-1/2 + sqrt(5)/2))/5 - sqrt(5)*log(-1/2 + sqrt(5)/2)/5 + sqrt(5)*(pi*i + log(1/2 + sqrt(5)/2))/5 + sqrt(5)*log(1/2 + sqrt(5)/2)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.