Sr Examen
Integral de 1/(x^2+4x+6) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | x + 4*x + 6 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 6}\, dx$$
Integral(1/(x^2 + 4*x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | x + 4*x + 6 | /
Reescribimos la función subintegral
1 1
------------ = ----------------------------
2 / 2 \
x + 4*x + 6 |/ ___ \ |
||-\/ 2 ___| |
2*||-------*x - \/ 2 | + 1|
\\ 2 / /o
/ | | 1 | ------------ dx | 2 = | x + 4*x + 6 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1
| \ 2 /
|
/
------------------------------
2 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1
| \ 2 /
|
/
------------------------------
2 hacemos el cambio
___
___ x*\/ 2
v = - \/ 2 - -------
2 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
2 2 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1 / ___\
| \ 2 / ___ | ___ x*\/ 2 |
| \/ 2 *atan|\/ 2 + -------|
/ \ 2 /
------------------------------ = ---------------------------
2 2 La solución:
/ ___\
___ | ___ x*\/ 2 |
\/ 2 *atan|\/ 2 + -------|
\ 2 /
C + ---------------------------
2
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 2 *(2 + x)| | \/ 2 *atan|-------------| | 1 \ 2 / | ------------ dx = C + ------------------------- | 2 2 | x + 4*x + 6 | /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 6}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 2\right)}{2} \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\
___ |3*\/ 2 |
\/ 2 *atan|-------| ___ / ___\
\ 2 / \/ 2 *atan\\/ 2 /
------------------- - -----------------
2 2
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2}$$
=
=
/ ___\
___ |3*\/ 2 |
\/ 2 *atan|-------| ___ / ___\
\ 2 / \/ 2 *atan\\/ 2 /
------------------- - -----------------
2 2
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2}$$
sqrt(2)*atan(3*sqrt(2)/2)/2 - sqrt(2)*atan(sqrt(2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.