Sr Examen
Integral de dx/(x^2-3*x+6) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | x - 3*x + 6 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - 3 x\right) + 6}\, dx$$
Integral(1/(x^2 - 3*x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | x - 3*x + 6 | /
Reescribimos la función subintegral
1 1
------------ = ----------------------------------
2 / 2 \
x - 3*x + 6 |/ ____ ____\ |
||-2*\/ 15 \/ 15 | |
15/4*||---------*x + ------| + 1|
\\ 15 5 / /o
/ | | 1 | ------------ dx | 2 = | x - 3*x + 6 | /
/
|
| 1
4* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x + ------| + 1
| \ 15 5 /
|
/
-----------------------------------
15 En integral
/
|
| 1
4* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x + ------| + 1
| \ 15 5 /
|
/
-----------------------------------
15 hacemos el cambio
____ ____
\/ 15 2*x*\/ 15
v = ------ - ----------
5 15 entonces
integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
15 15 hacemos cambio inverso
/
|
| 1
4* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x + ------| + 1 / ____ ____\
| \ 15 5 / ____ | \/ 15 2*x*\/ 15 |
| 2*\/ 15 *atan|- ------ + ----------|
/ \ 5 15 /
----------------------------------- = ------------------------------------
15 15 La solución:
/ ____ ____\
____ | \/ 15 2*x*\/ 15 |
2*\/ 15 *atan|- ------ + ----------|
\ 5 15 /
C + ------------------------------------
15
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 15 *(-3/2 + x)| | 2*\/ 15 *atan|-------------------| | 1 \ 15 / | ------------ dx = C + ---------------------------------- | 2 15 | x - 3*x + 6 | /
$$\int \frac{1}{\left(x^{2} - 3 x\right) + 6}\, dx = C + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{2 \sqrt{15} \left(x - \frac{3}{2}\right)}{15} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
2*\/ 15 *atan|------| 2*\/ 15 *atan|------|
\ 15 / \ 5 /
- --------------------- + ---------------------
15 15
$$- \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{15}$$
=
=
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
2*\/ 15 *atan|------| 2*\/ 15 *atan|------|
\ 15 / \ 5 /
- --------------------- + ---------------------
15 15
$$- \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{15}$$
-2*sqrt(15)*atan(sqrt(15)/15)/15 + 2*sqrt(15)*atan(sqrt(15)/5)/15
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.