Sr Examen
Integral de dx/(x^(2)+6x+11) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x + 6*x + 11 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 6 x\right) + 11}\, dx$$
Integral(1/(x^2 + 6*x + 11), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x + 6*x + 11 | /
Reescribimos la función subintegral
1 1
------------- = ------------------------------
2 / 2 \
x + 6*x + 11 |/ ___ ___\ |
||-\/ 2 3*\/ 2 | |
2*||-------*x - -------| + 1|
\\ 2 2 / /o
/ | | 1 | ------------- dx | 2 = | x + 6*x + 11 | /
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1
| \ 2 2 /
|
/
--------------------------------
2 En integral
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1
| \ 2 2 /
|
/
--------------------------------
2 hacemos el cambio
___ ___
3*\/ 2 x*\/ 2
v = - ------- - -------
2 2 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
2 2 hacemos cambio inverso
/
|
| 1
| -------------------------- dx
| 2
| / ___ ___\
| |-\/ 2 3*\/ 2 |
| |-------*x - -------| + 1 / ___ ___\
| \ 2 2 / ___ |3*\/ 2 x*\/ 2 |
| \/ 2 *atan|------- + -------|
/ \ 2 2 /
-------------------------------- = -----------------------------
2 2 La solución:
/ ___ ___\
___ |3*\/ 2 x*\/ 2 |
\/ 2 *atan|------- + -------|
\ 2 2 /
C + -----------------------------
2
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 2 *(3 + x)| | \/ 2 *atan|-------------| | 1 \ 2 / | ------------- dx = C + ------------------------- | 2 2 | x + 6*x + 11 | /
$$\int \frac{1}{\left(x^{2} + 6 x\right) + 11}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 3\right)}{2} \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\
___ |3*\/ 2 |
___ / ___\ \/ 2 *atan|-------|
\/ 2 *atan\2*\/ 2 / \ 2 /
------------------- - -------------------
2 2
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}$$
=
=
/ ___\
___ |3*\/ 2 |
___ / ___\ \/ 2 *atan|-------|
\/ 2 *atan\2*\/ 2 / \ 2 /
------------------- - -------------------
2 2
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(2 \sqrt{2} \right)}}{2}$$
sqrt(2)*atan(2*sqrt(2))/2 - sqrt(2)*atan(3*sqrt(2)/2)/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.