Sr Examen
Integral de (x^2+1)/(x^2+3x+1)^2 dx
Solución
Ha introducido
[src]
oo / | | 2 | x + 1 | --------------- dx | 2 | / 2 \ | \x + 3*x + 1/ | / 0
$$\int\limits_{0}^{\infty} \frac{x^{2} + 1}{\left(\left(x^{2} + 3 x\right) + 1\right)^{2}}\, dx$$
Integral((x^2 + 1)/(x^2 + 3*x + 1)^2, (x, 0, oo))
Respuesta (Indefinida)
[src]
// / ___ \ \
|| ___ |2*\/ 5 *(3/2 + x)| |
/ ||-\/ 5 *acoth|-----------------| | / ___\ / ___\
| || \ 5 / 2 | ___ |3 \/ 5 | ___ |3 \/ 5 |
| 2 ||-------------------------------- for (3/2 + x) > 5/4| 9*\/ 5 *log|- + x - -----| 9*\/ 5 *log|- + x + -----|
| x + 1 || 10 | 3*(2 + 3*x) \2 2 / \2 2 /
| --------------- dx = C + 4*|< | - --------------- - -------------------------- + --------------------------
| 2 || / ___ \ | 2 25 25
| / 2 \ || ___ |2*\/ 5 *(3/2 + x)| | 5 + 5*x + 15*x
| \x + 3*x + 1/ ||-\/ 5 *atanh|-----------------| |
| || \ 5 / 2 |
/ ||-------------------------------- for (3/2 + x) < 5/4|
\\ 10 /
$$\int \frac{x^{2} + 1}{\left(\left(x^{2} + 3 x\right) + 1\right)^{2}}\, dx = C - \frac{3 \left(3 x + 2\right)}{5 x^{2} + 15 x + 5} + 4 \left(\begin{cases} - \frac{\sqrt{5} \operatorname{acoth}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} > \frac{5}{4} \\- \frac{\sqrt{5} \operatorname{atanh}{\left(\frac{2 \sqrt{5} \left(x + \frac{3}{2}\right)}{5} \right)}}{10} & \text{for}\: \left(x + \frac{3}{2}\right)^{2} < \frac{5}{4} \end{cases}\right) - \frac{9 \sqrt{5} \log{\left(x - \frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{25} + \frac{9 \sqrt{5} \log{\left(x + \frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{25}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |3 \/ 5 | ___ |3 \/ 5 |
4*\/ 5 *log|- + -----| 4*\/ 5 *log|- - -----|
6 \2 2 / \2 2 /
- - ---------------------- + ----------------------
5 25 25
$$- \frac{4 \sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{25} + \frac{4 \sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{25} + \frac{6}{5}$$
=
=
/ ___\ / ___\
___ |3 \/ 5 | ___ |3 \/ 5 |
4*\/ 5 *log|- + -----| 4*\/ 5 *log|- - -----|
6 \2 2 / \2 2 /
- - ---------------------- + ----------------------
5 25 25
$$- \frac{4 \sqrt{5} \log{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}}{25} + \frac{4 \sqrt{5} \log{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}}{25} + \frac{6}{5}$$
6/5 - 4*sqrt(5)*log(3/2 + sqrt(5)/2)/25 + 4*sqrt(5)*log(3/2 - sqrt(5)/2)/25
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.