Sr Examen
Integral de (2*x+2)/(2*x^2+x+1) dx
Solución
Ha introducido
[src]
1 / | | 2*x + 2 | ------------ dx | 2 | 2*x + x + 1 | / 0
$$\int\limits_{0}^{1} \frac{2 x + 2}{\left(2 x^{2} + x\right) + 1}\, dx$$
Integral((2*x + 2)/(2*x^2 + x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x + 2 | ------------ dx | 2 | 2*x + x + 1 | /
Reescribimos la función subintegral
/ 2*2*x + 1 \
|------------| / 3 \
| 2 | |-----|
2*x + 2 \2*x + x + 1/ \2*7/8/
------------ = -------------- + -------------------------
2 2 2
2*x + x + 1 / ___ ___\
|-4*\/ 7 \/ 7 |
|--------*x - -----| + 1
\ 7 7 / o
/ | | 2*x + 2 | ------------ dx | 2 = | 2*x + x + 1 | /
/
|
| 1
/ 12* | ------------------------- dx
| | 2
| 2*2*x + 1 | / ___ ___\
| ------------ dx | |-4*\/ 7 \/ 7 |
| 2 | |--------*x - -----| + 1
| 2*x + x + 1 | \ 7 7 /
| |
/ /
------------------ + ----------------------------------
2 7 En integral
/
|
| 2*2*x + 1
| ------------ dx
| 2
| 2*x + x + 1
|
/
------------------
2 hacemos el cambio
2 u = x + 2*x
entonces
integral =
/
|
| 1
| ----- du
| 1 + u
|
/ log(1 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*2*x + 1
| ------------ dx
| 2
| 2*x + x + 1
| / 2\
/ log\1 + x + 2*x /
------------------ = -----------------
2 2 En integral
/
|
| 1
12* | ------------------------- dx
| 2
| / ___ ___\
| |-4*\/ 7 \/ 7 |
| |--------*x - -----| + 1
| \ 7 7 /
|
/
----------------------------------
7 hacemos el cambio
___ ___
\/ 7 4*x*\/ 7
v = - ----- - ---------
7 7 entonces
integral =
/
|
| 1
12* | ------ dv
| 2
| 1 + v
|
/ 12*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
12* | ------------------------- dx
| 2
| / ___ ___\
| |-4*\/ 7 \/ 7 |
| |--------*x - -----| + 1 / ___ ___\
| \ 7 7 / ___ |\/ 7 4*x*\/ 7 |
| 3*\/ 7 *atan|----- + ---------|
/ \ 7 7 /
---------------------------------- = -------------------------------
7 7 La solución:
/ ___ ___\
/1 2 x\ ___ |\/ 7 4*x*\/ 7 |
log|- + x + -| 3*\/ 7 *atan|----- + ---------|
\2 2/ \ 7 7 /
C + --------------- + -------------------------------
2 7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |4*\/ 7 *(1/4 + x)| | / 2\ 3*\/ 7 *atan|-----------------| | 2*x + 2 log\1 + x + 2*x / \ 7 / | ------------ dx = C + ----------------- + ------------------------------- | 2 2 7 | 2*x + x + 1 | /
$$\int \frac{2 x + 2}{\left(2 x^{2} + x\right) + 1}\, dx = C + \frac{\log{\left(2 x^{2} + x + 1 \right)}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{4 \sqrt{7} \left(x + \frac{1}{4}\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 7 | ___ |5*\/ 7 |
3*\/ 7 *atan|-----| 3*\/ 7 *atan|-------|
\ 7 / \ 7 /
- ------------------- + --------------------- + log(2)
7 7
$$- \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \log{\left(2 \right)} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7}$$
=
=
/ ___\ / ___\
___ |\/ 7 | ___ |5*\/ 7 |
3*\/ 7 *atan|-----| 3*\/ 7 *atan|-------|
\ 7 / \ 7 /
- ------------------- + --------------------- + log(2)
7 7
$$- \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \log{\left(2 \right)} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7}$$
-3*sqrt(7)*atan(sqrt(7)/7)/7 + 3*sqrt(7)*atan(5*sqrt(7)/7)/7 + log(2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.