Sr Examen
Integral de 1/(9*x^2-6*x+38) dx
Solución
Ha introducido
[src]
oo
/
|
| 1
| --------------- dx
| 2
| 9*x - 6*x + 38
|
/
___
4*\/ 7
-------
3
$$\int\limits_{\frac{4 \sqrt{7}}{3}}^{\infty} \frac{1}{\left(9 x^{2} - 6 x\right) + 38}\, dx$$
Integral(1/(9*x^2 - 6*x + 38), (x, 4*sqrt(7)/3, oo))
Solución detallada
Tenemos el integral:
/ | | 1 | --------------- dx | 2 | 9*x - 6*x + 38 | /
Reescribimos la función subintegral
1 1
--------------- = --------------------------------
2 / 2 \
9*x - 6*x + 38 |/ ____ ____\ |
||-3*\/ 37 \/ 37 | |
37*||---------*x + ------| + 1|
\\ 37 37 / /o
/ | | 1 | --------------- dx | 2 = | 9*x - 6*x + 38 | /
/
|
| 1
| --------------------------- dx
| 2
| / ____ ____\
| |-3*\/ 37 \/ 37 |
| |---------*x + ------| + 1
| \ 37 37 /
|
/
---------------------------------
37 En integral
/
|
| 1
| --------------------------- dx
| 2
| / ____ ____\
| |-3*\/ 37 \/ 37 |
| |---------*x + ------| + 1
| \ 37 37 /
|
/
---------------------------------
37 hacemos el cambio
____ ____
\/ 37 3*x*\/ 37
v = ------ - ----------
37 37 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
37 37 hacemos cambio inverso
/
|
| 1
| --------------------------- dx
| 2
| / ____ ____\
| |-3*\/ 37 \/ 37 |
| |---------*x + ------| + 1 / ____ ____\
| \ 37 37 / ____ | \/ 37 3*x*\/ 37 |
| \/ 37 *atan|- ------ + ----------|
/ \ 37 37 /
--------------------------------- = ----------------------------------
37 111 La solución:
/ ____ ____\
____ | \/ 37 3*x*\/ 37 |
\/ 37 *atan|- ------ + ----------|
\ 37 37 /
C + ----------------------------------
111
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |3*\/ 37 *(-1/3 + x)| | \/ 37 *atan|-------------------| | 1 \ 37 / | --------------- dx = C + -------------------------------- | 2 111 | 9*x - 6*x + 38 | /
$$\int \frac{1}{\left(9 x^{2} - 6 x\right) + 38}\, dx = C + \frac{\sqrt{37} \operatorname{atan}{\left(\frac{3 \sqrt{37} \left(x - \frac{1}{3}\right)}{37} \right)}}{111}$$
Gráfica
Respuesta
[src]
/ ____ _____\
____ | \/ 37 4*\/ 259 |
\/ 37 *atan|- ------ + ---------| ____
\ 37 37 / pi*\/ 37
- --------------------------------- + ---------
111 222
$$- \frac{\sqrt{37} \operatorname{atan}{\left(- \frac{\sqrt{37}}{37} + \frac{4 \sqrt{259}}{37} \right)}}{111} + \frac{\sqrt{37} \pi}{222}$$
=
=
/ ____ _____\
____ | \/ 37 4*\/ 259 |
\/ 37 *atan|- ------ + ---------| ____
\ 37 37 / pi*\/ 37
- --------------------------------- + ---------
111 222
$$- \frac{\sqrt{37} \operatorname{atan}{\left(- \frac{\sqrt{37}}{37} + \frac{4 \sqrt{259}}{37} \right)}}{111} + \frac{\sqrt{37} \pi}{222}$$
-sqrt(37)*atan(-sqrt(37)/37 + 4*sqrt(259)/37)/111 + pi*sqrt(37)/222
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.