Sr Examen
Integral de 1/(x^2+3*x+10) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x + 3*x + 10 | / -oo
$$\int\limits_{-\infty}^{1} \frac{1}{\left(x^{2} + 3 x\right) + 10}\, dx$$
Integral(1/(x^2 + 3*x + 10), (x, -oo, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x + 3*x + 10 | /
Reescribimos la función subintegral
1 1
------------- = ------------------------------------
2 / 2 \
x + 3*x + 10 |/ ____ ____\ |
||-2*\/ 31 3*\/ 31 | |
31/4*||---------*x - --------| + 1|
\\ 31 31 / /o
/ | | 1 | ------------- dx | 2 = | x + 3*x + 10 | /
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 31 3*\/ 31 |
| |---------*x - --------| + 1
| \ 31 31 /
|
/
-------------------------------------
31 En integral
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 31 3*\/ 31 |
| |---------*x - --------| + 1
| \ 31 31 /
|
/
-------------------------------------
31 hacemos el cambio
____ ____
3*\/ 31 2*x*\/ 31
v = - -------- - ----------
31 31 entonces
integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
31 31 hacemos cambio inverso
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 31 3*\/ 31 |
| |---------*x - --------| + 1 / ____ ____\
| \ 31 31 / ____ |3*\/ 31 2*x*\/ 31 |
| 2*\/ 31 *atan|-------- + ----------|
/ \ 31 31 /
------------------------------------- = ------------------------------------
31 31 La solución:
/ ____ ____\
____ |3*\/ 31 2*x*\/ 31 |
2*\/ 31 *atan|-------- + ----------|
\ 31 31 /
C + ------------------------------------
31
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 31 *(3/2 + x)| | 2*\/ 31 *atan|------------------| | 1 \ 31 / | ------------- dx = C + --------------------------------- | 2 31 | x + 3*x + 10 | /
$$\int \frac{1}{\left(x^{2} + 3 x\right) + 10}\, dx = C + \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{2 \sqrt{31} \left(x + \frac{3}{2}\right)}{31} \right)}}{31}$$
Gráfica
Respuesta
[src]
/ ____\
____ |5*\/ 31 |
____ 2*\/ 31 *atan|--------|
pi*\/ 31 \ 31 /
--------- + -----------------------
31 31
$$\frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{5 \sqrt{31}}{31} \right)}}{31} + \frac{\sqrt{31} \pi}{31}$$
=
=
/ ____\
____ |5*\/ 31 |
____ 2*\/ 31 *atan|--------|
pi*\/ 31 \ 31 /
--------- + -----------------------
31 31
$$\frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{5 \sqrt{31}}{31} \right)}}{31} + \frac{\sqrt{31} \pi}{31}$$
pi*sqrt(31)/31 + 2*sqrt(31)*atan(5*sqrt(31)/31)/31
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.