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