Sr Examen
Integral de 1/(7*x^(2)+3*x+1) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 7*x + 3*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(7 x^{2} + 3 x\right) + 1}\, dx$$
Integral(1/(7*x^2 + 3*x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 7*x + 3*x + 1 | /
Reescribimos la función subintegral
1 1
-------------- = -----------------------------------
2 / 2 \
7*x + 3*x + 1 |/ ____ ____\ |
19 ||-14*\/ 19 3*\/ 19 | |
--*||----------*x - --------| + 1|
28 \\ 19 19 / /o
/ | | 1 | -------------- dx | 2 = | 7*x + 3*x + 1 | /
/
|
| 1
28* | ------------------------------ dx
| 2
| / ____ ____\
| |-14*\/ 19 3*\/ 19 |
| |----------*x - --------| + 1
| \ 19 19 /
|
/
---------------------------------------
19 En integral
/
|
| 1
28* | ------------------------------ dx
| 2
| / ____ ____\
| |-14*\/ 19 3*\/ 19 |
| |----------*x - --------| + 1
| \ 19 19 /
|
/
---------------------------------------
19 hacemos el cambio
____ ____
3*\/ 19 14*x*\/ 19
v = - -------- - -----------
19 19 entonces
integral =
/
|
| 1
28* | ------ dv
| 2
| 1 + v
|
/ 28*atan(v)
--------------- = ----------
19 19 hacemos cambio inverso
/
|
| 1
28* | ------------------------------ dx
| 2
| / ____ ____\
| |-14*\/ 19 3*\/ 19 |
| |----------*x - --------| + 1 / ____ ____\
| \ 19 19 / ____ |3*\/ 19 14*x*\/ 19 |
| 2*\/ 19 *atan|-------- + -----------|
/ \ 19 19 /
--------------------------------------- = -------------------------------------
19 19 La solución:
/ ____ ____\
____ |3*\/ 19 14*x*\/ 19 |
2*\/ 19 *atan|-------- + -----------|
\ 19 19 /
C + -------------------------------------
19
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |14*\/ 19 *(3/14 + x)| | 2*\/ 19 *atan|--------------------| | 1 \ 19 / | -------------- dx = C + ----------------------------------- | 2 19 | 7*x + 3*x + 1 | /
$$\int \frac{1}{\left(7 x^{2} + 3 x\right) + 1}\, dx = C + \frac{2 \sqrt{19} \operatorname{atan}{\left(\frac{14 \sqrt{19} \left(x + \frac{3}{14}\right)}{19} \right)}}{19}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 19 | ____ |17*\/ 19 |
2*\/ 19 *atan|--------| 2*\/ 19 *atan|---------|
\ 19 / \ 19 /
- ----------------------- + ------------------------
19 19
$$- \frac{2 \sqrt{19} \operatorname{atan}{\left(\frac{3 \sqrt{19}}{19} \right)}}{19} + \frac{2 \sqrt{19} \operatorname{atan}{\left(\frac{17 \sqrt{19}}{19} \right)}}{19}$$
=
=
/ ____\ / ____\
____ |3*\/ 19 | ____ |17*\/ 19 |
2*\/ 19 *atan|--------| 2*\/ 19 *atan|---------|
\ 19 / \ 19 /
- ----------------------- + ------------------------
19 19
$$- \frac{2 \sqrt{19} \operatorname{atan}{\left(\frac{3 \sqrt{19}}{19} \right)}}{19} + \frac{2 \sqrt{19} \operatorname{atan}{\left(\frac{17 \sqrt{19}}{19} \right)}}{19}$$
-2*sqrt(19)*atan(3*sqrt(19)/19)/19 + 2*sqrt(19)*atan(17*sqrt(19)/19)/19
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.