Sr Examen
Integral de (x+1)/(4*x^2-12x+12) dx
Solución
Ha introducido
[src]
1 / | | x + 1 | ---------------- dx | 2 | 4*x - 12*x + 12 | / 0
$$\int\limits_{0}^{1} \frac{x + 1}{\left(4 x^{2} - 12 x\right) + 12}\, dx$$
Integral((x + 1)/(4*x^2 - 12*x + 12), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 1 | ---------------- dx | 2 | 4*x - 12*x + 12 | /
Reescribimos la función subintegral
/ 4*2*x - 12 \
|----------------| / 5 \
| 2 | |---|
x + 1 \4*x - 12*x + 12/ \2*3/
---------------- = ------------------ + -------------------------
2 8 2
4*x - 12*x + 12 / ___ \
|-2*\/ 3 ___|
|--------*x + \/ 3 | + 1
\ 3 / o
/ | | x + 1 | ---------------- dx | 2 = | 4*x - 12*x + 12 | /
/
|
| 1
/ 5* | ------------------------- dx
| | 2
| 4*2*x - 12 | / ___ \
| ---------------- dx | |-2*\/ 3 ___|
| 2 | |--------*x + \/ 3 | + 1
| 4*x - 12*x + 12 | \ 3 /
| |
/ /
---------------------- + ---------------------------------
8 6 En integral
/
|
| 4*2*x - 12
| ---------------- dx
| 2
| 4*x - 12*x + 12
|
/
----------------------
8 hacemos el cambio
2 u = -12*x + 4*x
entonces
integral =
/
|
| 1
| ------ du
| 12 + u
|
/ log(12 + u)
------------ = -----------
8 8 hacemos cambio inverso
/
|
| 4*2*x - 12
| ---------------- dx
| 2
| 4*x - 12*x + 12
| / 2 \
/ log\3 + x - 3*x/
---------------------- = -----------------
8 8 En integral
/
|
| 1
5* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 3 ___|
| |--------*x + \/ 3 | + 1
| \ 3 /
|
/
---------------------------------
6 hacemos el cambio
___
___ 2*x*\/ 3
v = \/ 3 - ---------
3 entonces
integral =
/
|
| 1
5* | ------ dv
| 2
| 1 + v
|
/ 5*atan(v)
-------------- = ---------
6 6 hacemos cambio inverso
/
|
| 1
5* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 3 ___|
| |--------*x + \/ 3 | + 1 / ___\
| \ 3 / ___ | ___ 2*x*\/ 3 |
| 5*\/ 3 *atan|- \/ 3 + ---------|
/ \ 3 /
--------------------------------- = ---------------------------------
6 12 La solución:
/ ___\
___ | ___ 2*x*\/ 3 |
/ 2 \ 5*\/ 3 *atan|- \/ 3 + ---------|
log\3 + x - 3*x/ \ 3 /
C + ----------------- + ---------------------------------
8 12
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 3 *(-3/2 + x)| | / 2 \ 5*\/ 3 *atan|------------------| | x + 1 log\3 + x - 3*x/ \ 3 / | ---------------- dx = C + ----------------- + -------------------------------- | 2 8 12 | 4*x - 12*x + 12 | /
$$\int \frac{x + 1}{\left(4 x^{2} - 12 x\right) + 12}\, dx = C + \frac{\log{\left(x^{2} - 3 x + 3 \right)}}{8} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{3}{2}\right)}{3} \right)}}{12}$$
Gráfica
Respuesta
[src]
___
log(3) 5*pi*\/ 3
- ------ + ----------
8 72
$$- \frac{\log{\left(3 \right)}}{8} + \frac{5 \sqrt{3} \pi}{72}$$
=
=
___
log(3) 5*pi*\/ 3
- ------ + ----------
8 72
$$- \frac{\log{\left(3 \right)}}{8} + \frac{5 \sqrt{3} \pi}{72}$$
-log(3)/8 + 5*pi*sqrt(3)/72
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.