Sr Examen
Integral de (x-1)/(6+4*x+x^2) dx
Solución
Ha introducido
[src]
1 / | | x - 1 | ------------ dx | 2 | 6 + 4*x + x | / 0
$$\int\limits_{0}^{1} \frac{x - 1}{x^{2} + \left(4 x + 6\right)}\, dx$$
Integral((x - 1)/(6 + 4*x + x^2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x - 1 | ------------ dx | 2 | 6 + 4*x + x | /
Reescribimos la función subintegral
/ 2*x + 4 \
|------------| /-3 \
| 2 | |---|
x - 1 \x + 4*x + 6/ \ 2 /
------------ = -------------- + ------------------------
2 2 2
6 + 4*x + x / ___ \
|-\/ 2 ___|
|-------*x - \/ 2 | + 1
\ 2 / o
/ | | x - 1 | ------------ dx | 2 = | 6 + 4*x + x | /
/
|
| 1
/ 3* | ------------------------ dx
| | 2
| 2*x + 4 | / ___ \
| ------------ dx | |-\/ 2 ___|
| 2 | |-------*x - \/ 2 | + 1
| x + 4*x + 6 | \ 2 /
| |
/ /
------------------ - --------------------------------
2 2 En integral
/
|
| 2*x + 4
| ------------ dx
| 2
| x + 4*x + 6
|
/
------------------
2 hacemos el cambio
2 u = x + 4*x
entonces
integral =
/
|
| 1
| ----- du
| 6 + u
|
/ log(6 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x + 4
| ------------ dx
| 2
| x + 4*x + 6
| / 2 \
/ log\6 + x + 4*x/
------------------ = -----------------
2 2 En integral
/
|
| 1
-3* | ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1
| \ 2 /
|
/
---------------------------------
2 hacemos el cambio
___
___ x*\/ 2
v = - \/ 2 - -------
2 entonces
integral =
/
|
| 1
-3* | ------ dv
| 2
| 1 + v
|
/ -3*atan(v)
--------------- = ----------
2 2 hacemos cambio inverso
/
|
| 1
-3* | ------------------------ dx
| 2
| / ___ \
| |-\/ 2 ___|
| |-------*x - \/ 2 | + 1 / ___\
| \ 2 / ___ | ___ x*\/ 2 |
| -3*\/ 2 *atan|\/ 2 + -------|
/ \ 2 /
--------------------------------- = ------------------------------
2 2 La solución:
/ ___\
___ | ___ x*\/ 2 |
/ 2 \ 3*\/ 2 *atan|\/ 2 + -------|
log\6 + x + 4*x/ \ 2 /
C + ----------------- - -----------------------------
2 2
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 2 *(2 + x)| | / 2 \ 3*\/ 2 *atan|-------------| | x - 1 log\6 + x + 4*x/ \ 2 / | ------------ dx = C + ----------------- - --------------------------- | 2 2 2 | 6 + 4*x + x | /
$$\int \frac{x - 1}{x^{2} + \left(4 x + 6\right)}\, dx = C + \frac{\log{\left(x^{2} + 4 x + 6 \right)}}{2} - \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 2\right)}{2} \right)}}{2}$$
Gráfica
Respuesta
[src]
/ ___\
___ |3*\/ 2 |
3*\/ 2 *atan|-------| ___ / ___\
log(11) log(6) \ 2 / 3*\/ 2 *atan\\/ 2 /
------- - ------ - --------------------- + -------------------
2 2 2 2
$$- \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} - \frac{\log{\left(6 \right)}}{2} + \frac{\log{\left(11 \right)}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
=
=
/ ___\
___ |3*\/ 2 |
3*\/ 2 *atan|-------| ___ / ___\
log(11) log(6) \ 2 / 3*\/ 2 *atan\\/ 2 /
------- - ------ - --------------------- + -------------------
2 2 2 2
$$- \frac{3 \sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{2} \right)}}{2} - \frac{\log{\left(6 \right)}}{2} + \frac{\log{\left(11 \right)}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}}{2}$$
log(11)/2 - log(6)/2 - 3*sqrt(2)*atan(3*sqrt(2)/2)/2 + 3*sqrt(2)*atan(sqrt(2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.