Sr Examen
Integral de (3x-1)/(3x^(2)+12x+1) dx
Solución
Ha introducido
[src]
1 / | | 3*x - 1 | --------------- dx | 2 | 3*x + 12*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{3 x - 1}{\left(3 x^{2} + 12 x\right) + 1}\, dx$$
Integral((3*x - 1)/(3*x^2 + 12*x + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ____ \ \
|| ____ |\/ 33 *(2 + x)| |
||-\/ 33 *acoth|--------------| |
/ || \ 11 / 2 |
| / 2 \ ||------------------------------ for (2 + x) > 11/3|
| 3*x - 1 log\1 + 3*x + 12*x/ || 33 |
| --------------- dx = C + -------------------- - 7*|< |
| 2 2 || / ____ \ |
| 3*x + 12*x + 1 || ____ |\/ 33 *(2 + x)| |
| ||-\/ 33 *atanh|--------------| |
/ || \ 11 / 2 |
||------------------------------ for (2 + x) < 11/3|
\\ 33 /
$$\int \frac{3 x - 1}{\left(3 x^{2} + 12 x\right) + 1}\, dx = C - 7 \left(\begin{cases} - \frac{\sqrt{33} \operatorname{acoth}{\left(\frac{\sqrt{33} \left(x + 2\right)}{11} \right)}}{33} & \text{for}\: \left(x + 2\right)^{2} > \frac{11}{3} \\- \frac{\sqrt{33} \operatorname{atanh}{\left(\frac{\sqrt{33} \left(x + 2\right)}{11} \right)}}{33} & \text{for}\: \left(x + 2\right)^{2} < \frac{11}{3} \end{cases}\right) + \frac{\log{\left(3 x^{2} + 12 x + 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |- - --------|*log|3 - ------| + |- + --------|*log|3 + ------| - |- - --------|*log|2 - ------| - |- + --------|*log|2 + ------| \2 66 / \ 3 / \2 66 / \ 3 / \2 66 / \ 3 / \2 66 / \ 3 /
$$- \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 2 \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(2 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(3 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 3 \right)}$$
=
=
/ ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ / ____\ |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |1 7*\/ 33 | | \/ 33 | |- - --------|*log|3 - ------| + |- + --------|*log|3 + ------| - |- - --------|*log|2 - ------| - |- + --------|*log|2 + ------| \2 66 / \ 3 / \2 66 / \ 3 / \2 66 / \ 3 / \2 66 / \ 3 /
$$- \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 2 \right)} - \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(2 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} - \frac{7 \sqrt{33}}{66}\right) \log{\left(3 - \frac{\sqrt{33}}{3} \right)} + \left(\frac{1}{2} + \frac{7 \sqrt{33}}{66}\right) \log{\left(\frac{\sqrt{33}}{3} + 3 \right)}$$
(1/2 - 7*sqrt(33)/66)*log(3 - sqrt(33)/3) + (1/2 + 7*sqrt(33)/66)*log(3 + sqrt(33)/3) - (1/2 - 7*sqrt(33)/66)*log(2 - sqrt(33)/3) - (1/2 + 7*sqrt(33)/66)*log(2 + sqrt(33)/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.