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