Sr Examen
Integral de 1/(x^2+3*x+11) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x + 3*x + 11 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 3 x\right) + 11}\, dx$$
Integral(1/(x^2 + 3*x + 11), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x + 3*x + 11 | /
Reescribimos la función subintegral
1 1
------------- = ------------------------------------
2 / 2 \
x + 3*x + 11 |/ ____ ____\ |
||-2*\/ 35 3*\/ 35 | |
35/4*||---------*x - --------| + 1|
\\ 35 35 / /o
/ | | 1 | ------------- dx | 2 = | x + 3*x + 11 | /
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 35 3*\/ 35 |
| |---------*x - --------| + 1
| \ 35 35 /
|
/
-------------------------------------
35 En integral
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 35 3*\/ 35 |
| |---------*x - --------| + 1
| \ 35 35 /
|
/
-------------------------------------
35 hacemos el cambio
____ ____
3*\/ 35 2*x*\/ 35
v = - -------- - ----------
35 35 entonces
integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
35 35 hacemos cambio inverso
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 35 3*\/ 35 |
| |---------*x - --------| + 1 / ____ ____\
| \ 35 35 / ____ |3*\/ 35 2*x*\/ 35 |
| 2*\/ 35 *atan|-------- + ----------|
/ \ 35 35 /
------------------------------------- = ------------------------------------
35 35 La solución:
/ ____ ____\
____ |3*\/ 35 2*x*\/ 35 |
2*\/ 35 *atan|-------- + ----------|
\ 35 35 /
C + ------------------------------------
35
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 35 *(3/2 + x)| | 2*\/ 35 *atan|------------------| | 1 \ 35 / | ------------- dx = C + --------------------------------- | 2 35 | x + 3*x + 11 | /
$$\int \frac{1}{\left(x^{2} + 3 x\right) + 11}\, dx = C + \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{2 \sqrt{35} \left(x + \frac{3}{2}\right)}{35} \right)}}{35}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 35 | ____ |\/ 35 |
2*\/ 35 *atan|--------| 2*\/ 35 *atan|------|
\ 35 / \ 7 /
- ----------------------- + ---------------------
35 35
$$- \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{3 \sqrt{35}}{35} \right)}}{35} + \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}}{35}$$
=
=
/ ____\ / ____\
____ |3*\/ 35 | ____ |\/ 35 |
2*\/ 35 *atan|--------| 2*\/ 35 *atan|------|
\ 35 / \ 7 /
- ----------------------- + ---------------------
35 35
$$- \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{3 \sqrt{35}}{35} \right)}}{35} + \frac{2 \sqrt{35} \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}}{35}$$
-2*sqrt(35)*atan(3*sqrt(35)/35)/35 + 2*sqrt(35)*atan(sqrt(35)/7)/35
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.