Sr Examen
Integral de dx/(2*x^2+x+2) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | 2*x + x + 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(2 x^{2} + x\right) + 2}\, dx$$
Integral(1/(2*x^2 + x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | 2*x + x + 2 | /
Reescribimos la función subintegral
1 1
------------ = ----------------------------------
2 / 2 \
2*x + x + 2 |/ ____ ____\ |
||-4*\/ 15 \/ 15 | |
15/8*||---------*x - ------| + 1|
\\ 15 15 / /o
/ | | 1 | ------------ dx | 2 = | 2*x + x + 2 | /
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 15 \/ 15 |
| |---------*x - ------| + 1
| \ 15 15 /
|
/
-----------------------------------
15 En integral
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 15 \/ 15 |
| |---------*x - ------| + 1
| \ 15 15 /
|
/
-----------------------------------
15 hacemos el cambio
____ ____
\/ 15 4*x*\/ 15
v = - ------ - ----------
15 15 entonces
integral =
/
|
| 1
8* | ------ dv
| 2
| 1 + v
|
/ 8*atan(v)
-------------- = ---------
15 15 hacemos cambio inverso
/
|
| 1
8* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 15 \/ 15 |
| |---------*x - ------| + 1 / ____ ____\
| \ 15 15 / ____ |\/ 15 4*x*\/ 15 |
| 2*\/ 15 *atan|------ + ----------|
/ \ 15 15 /
----------------------------------- = ----------------------------------
15 15 La solución:
/ ____ ____\
____ |\/ 15 4*x*\/ 15 |
2*\/ 15 *atan|------ + ----------|
\ 15 15 /
C + ----------------------------------
15
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |4*\/ 15 *(1/4 + x)| | 2*\/ 15 *atan|------------------| | 1 \ 15 / | ------------ dx = C + --------------------------------- | 2 15 | 2*x + x + 2 | /
$$\int \frac{1}{\left(2 x^{2} + x\right) + 2}\, dx = C + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{4 \sqrt{15} \left(x + \frac{1}{4}\right)}{15} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
2*\/ 15 *atan|------| 2*\/ 15 *atan|------|
\ 15 / \ 3 /
- --------------------- + ---------------------
15 15
$$- \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{3} \right)}}{15}$$
=
=
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
2*\/ 15 *atan|------| 2*\/ 15 *atan|------|
\ 15 / \ 3 /
- --------------------- + ---------------------
15 15
$$- \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{2 \sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{3} \right)}}{15}$$
-2*sqrt(15)*atan(sqrt(15)/15)/15 + 2*sqrt(15)*atan(sqrt(15)/3)/15
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.