Sr Examen
Integral de (2/3-1/3*x)/(x^2+x+1) dx
Solución
Ha introducido
[src]
1 / | | 2 x | - - - | 3 3 | ---------- dx | 2 | x + x + 1 | / 0
$$\int\limits_{0}^{1} \frac{\frac{2}{3} - \frac{x}{3}}{\left(x^{2} + x\right) + 1}\, dx$$
Integral((2/3 - x/3)/(x^2 + x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2 x | - - - | 3 3 | ---------- dx | 2 | x + x + 1 | /
Reescribimos la función subintegral
/ 2*x + 1 \
2 x |----------| / 5 \
- - - | 2 | |-----|
3 3 \x + x + 1/ \6*3/4/
---------- = - ------------ + -------------------------
2 6 2
x + x + 1 / ___ ___\
|-2*\/ 3 \/ 3 |
|--------*x - -----| + 1
\ 3 3 / o
/ | | 2 x | - - - | 3 3 | ---------- dx = | 2 | x + x + 1 | /
/
|
| 1
/ 10* | ------------------------- dx
| | 2
| 2*x + 1 | / ___ ___\
| ---------- dx | |-2*\/ 3 \/ 3 |
| 2 | |--------*x - -----| + 1
| x + x + 1 | \ 3 3 /
| |
/ /
- ---------------- + ----------------------------------
6 9 En integral
/
|
| 2*x + 1
- | ---------- dx
| 2
| x + x + 1
|
/
------------------
6 hacemos el cambio
2 u = x + x
entonces
integral =
/
|
| 1
- | ----- du
| 1 + u
|
/ -log(1 + u)
------------- = ------------
6 6 hacemos cambio inverso
/
|
| 2*x + 1
- | ---------- dx
| 2
| x + x + 1
| / 2\
/ -log\1 + x + x /
------------------ = -----------------
6 6 En integral
/
|
| 1
10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x - -----| + 1
| \ 3 3 /
|
/
----------------------------------
9 hacemos el cambio
___ ___
\/ 3 2*x*\/ 3
v = - ----- - ---------
3 3 entonces
integral =
/
|
| 1
10* | ------ dv
| 2
| 1 + v
|
/ 10*atan(v)
--------------- = ----------
9 9 hacemos cambio inverso
/
|
| 1
10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 \/ 3 |
| |--------*x - -----| + 1 / ___ ___\
| \ 3 3 / ___ |\/ 3 2*x*\/ 3 |
| 5*\/ 3 *atan|----- + ---------|
/ \ 3 3 /
---------------------------------- = -------------------------------
9 9 La solución:
/ ___ ___\
___ |\/ 3 2*x*\/ 3 |
/ 2\ 5*\/ 3 *atan|----- + ---------|
log\1 + x + x / \ 3 3 /
C - --------------- + -------------------------------
6 9
Respuesta (Indefinida)
[src]
/ | / ___ \ | 2 x ___ |2*\/ 3 *(1/2 + x)| | - - - / 2\ 5*\/ 3 *atan|-----------------| | 3 3 log\1 + x + x / \ 3 / | ---------- dx = C - --------------- + ------------------------------- | 2 6 9 | x + x + 1 | /
$$\int \frac{\frac{2}{3} - \frac{x}{3}}{\left(x^{2} + x\right) + 1}\, dx = C - \frac{\log{\left(x^{2} + x + 1 \right)}}{6} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x + \frac{1}{2}\right)}{3} \right)}}{9}$$
Gráfica
Respuesta
[src]
___
log(3) 5*pi*\/ 3
- ------ + ----------
6 54
$$- \frac{\log{\left(3 \right)}}{6} + \frac{5 \sqrt{3} \pi}{54}$$
=
=
___
log(3) 5*pi*\/ 3
- ------ + ----------
6 54
$$- \frac{\log{\left(3 \right)}}{6} + \frac{5 \sqrt{3} \pi}{54}$$
-log(3)/6 + 5*pi*sqrt(3)/54
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.