Sr Examen
Integral de (x+2)/(x^2+3*x+1) dx
Solución
Ha introducido
[src]
1 / | | x + 2 | ------------ dx | 2 | x + 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{x + 2}{\left(x^{2} + 3 x\right) + 1}\, dx$$
Integral((x + 2)/(x^2 + 3*x + 1), (x, 0, 1))
Respuesta (Indefinida)
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// / ___ \ \
|| ___ |2*\/ 5 *(3/2 + x)| |
||-\/ 5 *acoth|-----------------| |
/ || \ 5 / 2 |
| / 2 \ ||-------------------------------- for (3/2 + x) > 5/4|
| x + 2 log\1 + x + 3*x/ || 10 |
| ------------ dx = C + ----------------- + 2*|< |
| 2 2 || / ___ \ |
| x + 3*x + 1 || ___ |2*\/ 5 *(3/2 + x)| |
| ||-\/ 5 *atanh|-----------------| |
/ || \ 5 / 2 |
||-------------------------------- for (3/2 + x) < 5/4|
\\ 10 /
$$\int \frac{x + 2}{\left(x^{2} + 3 x\right) + 1}\, dx = C + 2 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) + \frac{\log{\left(x^{2} + 3 x + 1 \right)}}{2}$$
Gráfica
Respuesta
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/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ |1 \/ 5 | |5 \/ 5 | |1 \/ 5 | |5 \/ 5 | |1 \/ 5 | |3 \/ 5 | |1 \/ 5 | |3 \/ 5 | |- - -----|*log|- + -----| + |- + -----|*log|- - -----| - |- - -----|*log|- + -----| - |- + -----|*log|- - -----| \2 10 / \2 2 / \2 10 / \2 2 / \2 10 / \2 2 / \2 10 / \2 2 /
$$- \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}$$
=
=
/ ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ / ___\ |1 \/ 5 | |5 \/ 5 | |1 \/ 5 | |5 \/ 5 | |1 \/ 5 | |3 \/ 5 | |1 \/ 5 | |3 \/ 5 | |- - -----|*log|- + -----| + |- + -----|*log|- - -----| - |- - -----|*log|- + -----| - |- + -----|*log|- - -----| \2 10 / \2 2 / \2 10 / \2 2 / \2 10 / \2 2 / \2 10 / \2 2 /
$$- \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} + \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{5}{2} - \frac{\sqrt{5}}{2} \right)} + \left(\frac{1}{2} - \frac{\sqrt{5}}{10}\right) \log{\left(\frac{\sqrt{5}}{2} + \frac{5}{2} \right)} - \left(\frac{\sqrt{5}}{10} + \frac{1}{2}\right) \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}$$
(1/2 - sqrt(5)/10)*log(5/2 + sqrt(5)/2) + (1/2 + sqrt(5)/10)*log(5/2 - sqrt(5)/2) - (1/2 - sqrt(5)/10)*log(3/2 + sqrt(5)/2) - (1/2 + sqrt(5)/10)*log(3/2 - sqrt(5)/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.