Sr Examen
Integral de 1/(6x^2-3x+2) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 6*x - 3*x + 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(6 x^{2} - 3 x\right) + 2}\, dx$$
Integral(1/(6*x^2 - 3*x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 6*x - 3*x + 2 | /
Reescribimos la función subintegral
1 1
-------------- = ----------------------------------
2 / 2 \
6*x - 3*x + 2 |/ ____ ____\ |
||-4*\/ 39 \/ 39 | |
13/8*||---------*x + ------| + 1|
\\ 13 13 / /o
/ | | 1 | -------------- dx | 2 = | 6*x - 3*x + 2 | /
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1
| \ 13 13 /
|
/
-----------------------------------
13 En integral
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1
| \ 13 13 /
|
/
-----------------------------------
13 hacemos el cambio
____ ____
\/ 39 4*x*\/ 39
v = ------ - ----------
13 13 entonces
integral =
/
|
| 1
8* | ------ dv
| 2
| 1 + v
|
/ 8*atan(v)
-------------- = ---------
13 13 hacemos cambio inverso
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x + ------| + 1 / ____ ____\
| \ 13 13 / ____ | \/ 39 4*x*\/ 39 |
| 2*\/ 39 *atan|- ------ + ----------|
/ \ 13 13 /
----------------------------------- = ------------------------------------
13 39 La solución:
/ ____ ____\
____ | \/ 39 4*x*\/ 39 |
2*\/ 39 *atan|- ------ + ----------|
\ 13 13 /
C + ------------------------------------
39
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |4*\/ 39 *(-1/4 + x)| | 2*\/ 39 *atan|-------------------| | 1 \ 13 / | -------------- dx = C + ---------------------------------- | 2 39 | 6*x - 3*x + 2 | /
$$\int \frac{1}{\left(6 x^{2} - 3 x\right) + 2}\, dx = C + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{4 \sqrt{39} \left(x - \frac{1}{4}\right)}{13} \right)}}{39}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 13 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39}$$
=
=
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 13 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39}$$
2*sqrt(39)*atan(sqrt(39)/13)/39 + 2*sqrt(39)*atan(3*sqrt(39)/13)/39
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.