Sr Examen
Integral de (4x-3)/(1+5x-4x^2) dx
Solución
Ha introducido
[src]
1 / | | 4*x - 3 | -------------- dx | 2 | 1 + 5*x - 4*x | / 0
$$\int\limits_{0}^{1} \frac{4 x - 3}{- 4 x^{2} + \left(5 x + 1\right)}\, dx$$
Integral((4*x - 3)/(1 + 5*x - 4*x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ____ \ \
|| ____ |8*\/ 41 *(-5/8 + x)| |
||-\/ 41 *acoth|-------------------| |
/ || \ 41 / 2 41|
| ||----------------------------------- for (-5/8 + x) > --| / 2\
| 4*x - 3 || 328 64| log\-1 - 5*x + 4*x /
| -------------- dx = C + 8*|< | - --------------------
| 2 || / ____ \ | 2
| 1 + 5*x - 4*x || ____ |8*\/ 41 *(-5/8 + x)| |
| ||-\/ 41 *atanh|-------------------| |
/ || \ 41 / 2 41|
||----------------------------------- for (-5/8 + x) < --|
\\ 328 64/
$$\int \frac{4 x - 3}{- 4 x^{2} + \left(5 x + 1\right)}\, dx = C + 8 \left(\begin{cases} - \frac{\sqrt{41} \operatorname{acoth}{\left(\frac{8 \sqrt{41} \left(x - \frac{5}{8}\right)}{41} \right)}}{328} & \text{for}\: \left(x - \frac{5}{8}\right)^{2} > \frac{41}{64} \\- \frac{\sqrt{41} \operatorname{atanh}{\left(\frac{8 \sqrt{41} \left(x - \frac{5}{8}\right)}{41} \right)}}{328} & \text{for}\: \left(x - \frac{5}{8}\right)^{2} < \frac{41}{64} \end{cases}\right) - \frac{\log{\left(4 x^{2} - 5 x - 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ |1 \/ 41 | | |5 \/ 41 || |1 \/ 41 | | 5 \/ 41 | |1 \/ 41 | | | 3 \/ 41 || |1 \/ 41 | |3 \/ 41 | |- - ------|*|pi*I + log|- + ------|| + |- + ------|*log|- - + ------| - |- - ------|*|pi*I + log|- - + ------|| - |- + ------|*log|- + ------| \2 82 / \ \8 8 // \2 82 / \ 8 8 / \2 82 / \ \ 8 8 // \2 82 / \8 8 /
$$\left(\frac{\sqrt{41}}{82} + \frac{1}{2}\right) \log{\left(- \frac{5}{8} + \frac{\sqrt{41}}{8} \right)} - \left(\frac{\sqrt{41}}{82} + \frac{1}{2}\right) \log{\left(\frac{3}{8} + \frac{\sqrt{41}}{8} \right)} - \left(\frac{1}{2} - \frac{\sqrt{41}}{82}\right) \left(\log{\left(- \frac{3}{8} + \frac{\sqrt{41}}{8} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{41}}{82}\right) \left(\log{\left(\frac{5}{8} + \frac{\sqrt{41}}{8} \right)} + i \pi\right)$$
=
=
/ ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ |1 \/ 41 | | |5 \/ 41 || |1 \/ 41 | | 5 \/ 41 | |1 \/ 41 | | | 3 \/ 41 || |1 \/ 41 | |3 \/ 41 | |- - ------|*|pi*I + log|- + ------|| + |- + ------|*log|- - + ------| - |- - ------|*|pi*I + log|- - + ------|| - |- + ------|*log|- + ------| \2 82 / \ \8 8 // \2 82 / \ 8 8 / \2 82 / \ \ 8 8 // \2 82 / \8 8 /
$$\left(\frac{\sqrt{41}}{82} + \frac{1}{2}\right) \log{\left(- \frac{5}{8} + \frac{\sqrt{41}}{8} \right)} - \left(\frac{\sqrt{41}}{82} + \frac{1}{2}\right) \log{\left(\frac{3}{8} + \frac{\sqrt{41}}{8} \right)} - \left(\frac{1}{2} - \frac{\sqrt{41}}{82}\right) \left(\log{\left(- \frac{3}{8} + \frac{\sqrt{41}}{8} \right)} + i \pi\right) + \left(\frac{1}{2} - \frac{\sqrt{41}}{82}\right) \left(\log{\left(\frac{5}{8} + \frac{\sqrt{41}}{8} \right)} + i \pi\right)$$
(1/2 - sqrt(41)/82)*(pi*i + log(5/8 + sqrt(41)/8)) + (1/2 + sqrt(41)/82)*log(-5/8 + sqrt(41)/8) - (1/2 - sqrt(41)/82)*(pi*i + log(-3/8 + sqrt(41)/8)) - (1/2 + sqrt(41)/82)*log(3/8 + sqrt(41)/8)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.