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