Sr Examen
Integral de (3x-2)/(x^2+5x+1) dx
Solución
Ha introducido
[src]
1 / | | 3*x - 2 | ------------ dx | 2 | x + 5*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{3 x - 2}{\left(x^{2} + 5 x\right) + 1}\, dx$$
Integral((3*x - 2)/(x^2 + 5*x + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ____ \ \
|| ____ |2*\/ 21 *(5/2 + x)| |
||-\/ 21 *acoth|------------------| |
/ || \ 21 / 2 |
| ||---------------------------------- for (5/2 + x) > 21/4| / 2 \
| 3*x - 2 || 42 | 3*log\1 + x + 5*x/
| ------------ dx = C - 38*|< | + -------------------
| 2 || / ____ \ | 2
| x + 5*x + 1 || ____ |2*\/ 21 *(5/2 + x)| |
| ||-\/ 21 *atanh|------------------| |
/ || \ 21 / 2 |
||---------------------------------- for (5/2 + x) < 21/4|
\\ 42 /
$$\int \frac{3 x - 2}{\left(x^{2} + 5 x\right) + 1}\, dx = C - 38 \left(\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} > \frac{21}{4} \\- \frac{\sqrt{21} \operatorname{atanh}{\left(\frac{2 \sqrt{21} \left(x + \frac{5}{2}\right)}{21} \right)}}{42} & \text{for}\: \left(x + \frac{5}{2}\right)^{2} < \frac{21}{4} \end{cases}\right) + \frac{3 \log{\left(x^{2} + 5 x + 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ |3 19*\/ 21 | |7 \/ 21 | |3 19*\/ 21 | |7 \/ 21 | |3 19*\/ 21 | |5 \/ 21 | |3 19*\/ 21 | |5 \/ 21 | |- - ---------|*log|- - ------| + |- + ---------|*log|- + ------| - |- - ---------|*log|- - ------| - |- + ---------|*log|- + ------| \2 42 / \2 2 / \2 42 / \2 2 / \2 42 / \2 2 / \2 42 / \2 2 /
$$- \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)} - \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)}$$
=
=
/ ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ |3 19*\/ 21 | |7 \/ 21 | |3 19*\/ 21 | |7 \/ 21 | |3 19*\/ 21 | |5 \/ 21 | |3 19*\/ 21 | |5 \/ 21 | |- - ---------|*log|- - ------| + |- + ---------|*log|- + ------| - |- - ---------|*log|- - ------| - |- + ---------|*log|- + ------| \2 42 / \2 2 / \2 42 / \2 2 / \2 42 / \2 2 / \2 42 / \2 2 /
$$- \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)} - \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} - \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{7}{2} - \frac{\sqrt{21}}{2} \right)} + \left(\frac{3}{2} + \frac{19 \sqrt{21}}{42}\right) \log{\left(\frac{\sqrt{21}}{2} + \frac{7}{2} \right)}$$
(3/2 - 19*sqrt(21)/42)*log(7/2 - sqrt(21)/2) + (3/2 + 19*sqrt(21)/42)*log(7/2 + sqrt(21)/2) - (3/2 - 19*sqrt(21)/42)*log(5/2 - sqrt(21)/2) - (3/2 + 19*sqrt(21)/42)*log(5/2 + sqrt(21)/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.