Sr Examen
Integral de (3*x+1)/(x^2+4*x+1) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 1 | ------------ dx | 2 | x + 4*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 1}{\left(x^{2} + 4 x\right) + 1}\, dx$$
Integral((3*x + 1)/(x^2 + 4*x + 1), (x, 0, 1))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |\/ 3 *(2 + x)| |
||-\/ 3 *acoth|-------------| |
/ || \ 3 / 2 |
| ||---------------------------- for (2 + x) > 3| / 2 \
| 3*x + 1 || 3 | 3*log\1 + x + 4*x/
| ------------ dx = C - 5*|< | + -------------------
| 2 || / ___ \ | 2
| x + 4*x + 1 || ___ |\/ 3 *(2 + x)| |
| ||-\/ 3 *atanh|-------------| |
/ || \ 3 / 2 |
||---------------------------- for (2 + x) < 3|
\\ 3 /
$$\int \frac{3 x + 1}{\left(x^{2} + 4 x\right) + 1}\, dx = C - 5 \left(\begin{cases} - \frac{\sqrt{3} \operatorname{acoth}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3} & \text{for}\: \left(x + 2\right)^{2} > 3 \\- \frac{\sqrt{3} \operatorname{atanh}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3} & \text{for}\: \left(x + 2\right)^{2} < 3 \end{cases}\right) + \frac{3 \log{\left(x^{2} + 4 x + 1 \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\ / ___\ / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |- - -------|*log\3 - \/ 3 / + |- + -------|*log\3 + \/ 3 / - |- - -------|*log\2 - \/ 3 / - |- + -------|*log\2 + \/ 3 / \2 6 / \2 6 / \2 6 / \2 6 /
$$- \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 2 \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(3 - \sqrt{3} \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 3 \right)}$$
=
=
/ ___\ / ___\ / ___\ / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |3 5*\/ 3 | / ___\ |- - -------|*log\3 - \/ 3 / + |- + -------|*log\3 + \/ 3 / - |- - -------|*log\2 - \/ 3 / - |- + -------|*log\2 + \/ 3 / \2 6 / \2 6 / \2 6 / \2 6 /
$$- \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 2 \right)} + \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(3 - \sqrt{3} \right)} - \left(\frac{3}{2} - \frac{5 \sqrt{3}}{6}\right) \log{\left(2 - \sqrt{3} \right)} + \left(\frac{5 \sqrt{3}}{6} + \frac{3}{2}\right) \log{\left(\sqrt{3} + 3 \right)}$$
(3/2 - 5*sqrt(3)/6)*log(3 - sqrt(3)) + (3/2 + 5*sqrt(3)/6)*log(3 + sqrt(3)) - (3/2 - 5*sqrt(3)/6)*log(2 - sqrt(3)) - (3/2 + 5*sqrt(3)/6)*log(2 + sqrt(3))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.