Sr Examen
Integral de dx/(3*x^2-x+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | 3*x - x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 x^{2} - x\right) + 1}\, dx$$
Integral(1/(3*x^2 - x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | 3*x - x + 1 | /
Reescribimos la función subintegral
1 1
------------ = --------------------------------
2 / 2 \
3*x - x + 1 |/ ____ ____\ |
11 ||-6*\/ 11 \/ 11 | |
--*||---------*x + ------| + 1|
12 \\ 11 11 / /o
/ | | 1 | ------------ dx | 2 = | 3*x - x + 1 | /
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1
| \ 11 11 /
|
/
------------------------------------
11 En integral
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1
| \ 11 11 /
|
/
------------------------------------
11 hacemos el cambio
____ ____
\/ 11 6*x*\/ 11
v = ------ - ----------
11 11 entonces
integral =
/
|
| 1
12* | ------ dv
| 2
| 1 + v
|
/ 12*atan(v)
--------------- = ----------
11 11 hacemos cambio inverso
/
|
| 1
12* | --------------------------- dx
| 2
| / ____ ____\
| |-6*\/ 11 \/ 11 |
| |---------*x + ------| + 1 / ____ ____\
| \ 11 11 / ____ | \/ 11 6*x*\/ 11 |
| 2*\/ 11 *atan|- ------ + ----------|
/ \ 11 11 /
------------------------------------ = ------------------------------------
11 11 La solución:
/ ____ ____\
____ | \/ 11 6*x*\/ 11 |
2*\/ 11 *atan|- ------ + ----------|
\ 11 11 /
C + ------------------------------------
11
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |6*\/ 11 *(-1/6 + x)| | 2*\/ 11 *atan|-------------------| | 1 \ 11 / | ------------ dx = C + ---------------------------------- | 2 11 | 3*x - x + 1 | /
$$\int \frac{1}{\left(3 x^{2} - x\right) + 1}\, dx = C + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{6 \sqrt{11} \left(x - \frac{1}{6}\right)}{11} \right)}}{11}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 11 | ____ |5*\/ 11 |
2*\/ 11 *atan|------| 2*\/ 11 *atan|--------|
\ 11 / \ 11 /
--------------------- + -----------------------
11 11
$$\frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{5 \sqrt{11}}{11} \right)}}{11}$$
=
=
/ ____\ / ____\
____ |\/ 11 | ____ |5*\/ 11 |
2*\/ 11 *atan|------| 2*\/ 11 *atan|--------|
\ 11 / \ 11 /
--------------------- + -----------------------
11 11
$$\frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{5 \sqrt{11}}{11} \right)}}{11}$$
2*sqrt(11)*atan(sqrt(11)/11)/11 + 2*sqrt(11)*atan(5*sqrt(11)/11)/11
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.