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