Sr Examen
Integral de dx/(6*x^2-x+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | 6*x - x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(6 x^{2} - x\right) + 1}\, dx$$
Integral(1/(6*x^2 - x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | 6*x - x + 1 | /
Reescribimos la función subintegral
1 1
------------ = ---------------------------------
2 / 2 \
6*x - x + 1 |/ ____ ____\ |
23 ||-12*\/ 23 \/ 23 | |
--*||----------*x + ------| + 1|
24 \\ 23 23 / /o
/ | | 1 | ------------ dx | 2 = | 6*x - x + 1 | /
/
|
| 1
24* | ---------------------------- dx
| 2
| / ____ ____\
| |-12*\/ 23 \/ 23 |
| |----------*x + ------| + 1
| \ 23 23 /
|
/
-------------------------------------
23 En integral
/
|
| 1
24* | ---------------------------- dx
| 2
| / ____ ____\
| |-12*\/ 23 \/ 23 |
| |----------*x + ------| + 1
| \ 23 23 /
|
/
-------------------------------------
23 hacemos el cambio
____ ____
\/ 23 12*x*\/ 23
v = ------ - -----------
23 23 entonces
integral =
/
|
| 1
24* | ------ dv
| 2
| 1 + v
|
/ 24*atan(v)
--------------- = ----------
23 23 hacemos cambio inverso
/
|
| 1
24* | ---------------------------- dx
| 2
| / ____ ____\
| |-12*\/ 23 \/ 23 |
| |----------*x + ------| + 1 / ____ ____\
| \ 23 23 / ____ | \/ 23 12*x*\/ 23 |
| 2*\/ 23 *atan|- ------ + -----------|
/ \ 23 23 /
------------------------------------- = -------------------------------------
23 23 La solución:
/ ____ ____\
____ | \/ 23 12*x*\/ 23 |
2*\/ 23 *atan|- ------ + -----------|
\ 23 23 /
C + -------------------------------------
23
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |12*\/ 23 *(-1/12 + x)| | 2*\/ 23 *atan|---------------------| | 1 \ 23 / | ------------ dx = C + ------------------------------------ | 2 23 | 6*x - x + 1 | /
$$\int \frac{1}{\left(6 x^{2} - x\right) + 1}\, dx = C + \frac{2 \sqrt{23} \operatorname{atan}{\left(\frac{12 \sqrt{23} \left(x - \frac{1}{12}\right)}{23} \right)}}{23}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 23 | ____ |11*\/ 23 |
2*\/ 23 *atan|------| 2*\/ 23 *atan|---------|
\ 23 / \ 23 /
--------------------- + ------------------------
23 23
$$\frac{2 \sqrt{23} \operatorname{atan}{\left(\frac{\sqrt{23}}{23} \right)}}{23} + \frac{2 \sqrt{23} \operatorname{atan}{\left(\frac{11 \sqrt{23}}{23} \right)}}{23}$$
=
=
/ ____\ / ____\
____ |\/ 23 | ____ |11*\/ 23 |
2*\/ 23 *atan|------| 2*\/ 23 *atan|---------|
\ 23 / \ 23 /
--------------------- + ------------------------
23 23
$$\frac{2 \sqrt{23} \operatorname{atan}{\left(\frac{\sqrt{23}}{23} \right)}}{23} + \frac{2 \sqrt{23} \operatorname{atan}{\left(\frac{11 \sqrt{23}}{23} \right)}}{23}$$
2*sqrt(23)*atan(sqrt(23)/23)/23 + 2*sqrt(23)*atan(11*sqrt(23)/23)/23
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.