Sr Examen
Integral de (4+2x)/(4-x-2x²) dx
Solución
Ha introducido
[src]
1 / | | 4 + 2*x | ------------ dx | 2 | 4 - x - 2*x | / 0
$$\int\limits_{0}^{1} \frac{2 x + 4}{- 2 x^{2} + \left(4 - x\right)}\, dx$$
Integral((4 + 2*x)/(4 - x - 2*x^2), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ____ \ \
|| ____ |4*\/ 33 *(1/4 + x)| |
||-\/ 33 *acoth|------------------| |
/ || \ 33 / 2 33|
| ||---------------------------------- for (1/4 + x) > --| / 2\
| 4 + 2*x || 132 16| log\4 - x - 2*x /
| ------------ dx = C - 28*|< | - -----------------
| 2 || / ____ \ | 2
| 4 - x - 2*x || ____ |4*\/ 33 *(1/4 + x)| |
| ||-\/ 33 *atanh|------------------| |
/ || \ 33 / 2 33|
||---------------------------------- for (1/4 + x) < --|
\\ 132 16/
$$\int \frac{2 x + 4}{- 2 x^{2} + \left(4 - x\right)}\, dx = C - 28 \left(\begin{cases} - \frac{\sqrt{33} \operatorname{acoth}{\left(\frac{4 \sqrt{33} \left(x + \frac{1}{4}\right)}{33} \right)}}{132} & \text{for}\: \left(x + \frac{1}{4}\right)^{2} > \frac{33}{16} \\- \frac{\sqrt{33} \operatorname{atanh}{\left(\frac{4 \sqrt{33} \left(x + \frac{1}{4}\right)}{33} \right)}}{132} & \text{for}\: \left(x + \frac{1}{4}\right)^{2} < \frac{33}{16} \end{cases}\right) - \frac{\log{\left(- 2 x^{2} - x + 4 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ |1 7*\/ 33 | |1 \/ 33 | |1 7*\/ 33 | | | 1 \/ 33 || |1 7*\/ 33 | |5 \/ 33 | |1 7*\/ 33 | | | 5 \/ 33 || |- - --------|*log|- + ------| + |- + --------|*|pi*I + log|- - + ------|| - |- - --------|*log|- + ------| - |- + --------|*|pi*I + log|- - + ------|| \2 66 / \4 4 / \2 66 / \ \ 4 4 // \2 66 / \4 4 / \2 66 / \ \ 4 4 //
$$\left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{1}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{5}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{5}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{1}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right)$$
=
=
/ ____\ / ____\ / ____\ / / ____\\ / ____\ / ____\ / ____\ / / ____\\ |1 7*\/ 33 | |1 \/ 33 | |1 7*\/ 33 | | | 1 \/ 33 || |1 7*\/ 33 | |5 \/ 33 | |1 7*\/ 33 | | | 5 \/ 33 || |- - --------|*log|- + ------| + |- + --------|*|pi*I + log|- - + ------|| - |- - --------|*log|- + ------| - |- + --------|*|pi*I + log|- - + ------|| \2 66 / \4 4 / \2 66 / \ \ 4 4 // \2 66 / \4 4 / \2 66 / \ \ 4 4 //
$$\left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{1}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{5}{4} + \frac{\sqrt{33}}{4} \right)} - \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{5}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right) + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \left(\log{\left(- \frac{1}{4} + \frac{\sqrt{33}}{4} \right)} + i \pi\right)$$
(1/2 - 7*sqrt(33)/66)*log(1/4 + sqrt(33)/4) + (1/2 + 7*sqrt(33)/66)*(pi*i + log(-1/4 + sqrt(33)/4)) - (1/2 - 7*sqrt(33)/66)*log(5/4 + sqrt(33)/4) - (1/2 + 7*sqrt(33)/66)*(pi*i + log(-5/4 + sqrt(33)/4))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.