Sr Examen
Integral de (3x+2)/(x^2-4x+9) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 2 | ------------ dx | 2 | x - 4*x + 9 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 2}{\left(x^{2} - 4 x\right) + 9}\, dx$$
Integral((3*x + 2)/(x^2 - 4*x + 9), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x + 2 | ------------ dx | 2 | x - 4*x + 9 | /
Reescribimos la función subintegral
2*x - 4
3*------------ /8\
2 |-|
3*x + 2 x - 4*x + 9 \5/
------------ = -------------- + --------------------------
2 2 2
x - 4*x + 9 / ___ ___\
|-\/ 5 2*\/ 5 |
|-------*x + -------| + 1
\ 5 5 / o
/ | | 3*x + 2 | ------------ dx | 2 = | x - 4*x + 9 | /
/
|
| 1
/ 8* | -------------------------- dx
| | 2
| 2*x - 4 | / ___ ___\
3* | ------------ dx | |-\/ 5 2*\/ 5 |
| 2 | |-------*x + -------| + 1
| x - 4*x + 9 | \ 5 5 /
| |
/ /
-------------------- + ----------------------------------
2 5 En integral
/
|
| 2*x - 4
3* | ------------ dx
| 2
| x - 4*x + 9
|
/
--------------------
2 hacemos el cambio
2 u = x - 4*x
entonces
integral =
/
|
| 1
3* | ----- du
| 9 + u
|
/ 3*log(9 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x - 4
3* | ------------ dx
| 2
| x - 4*x + 9
| / 2 \
/ 3*log\9 + x - 4*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
8* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 5 2*\/ 5 |
| |-------*x + -------| + 1
| \ 5 5 /
|
/
----------------------------------
5 hacemos el cambio
___ ___
2*\/ 5 x*\/ 5
v = ------- - -------
5 5 entonces
integral =
/
|
| 1
8* | ------ dv
| 2
| 1 + v
|
/ 8*atan(v)
-------------- = ---------
5 5 hacemos cambio inverso
/
|
| 1
8* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 5 2*\/ 5 |
| |-------*x + -------| + 1 / ___ ___\
| \ 5 5 / ___ | 2*\/ 5 x*\/ 5 |
| 8*\/ 5 *atan|- ------- + -------|
/ \ 5 5 /
---------------------------------- = ---------------------------------
5 5 La solución:
/ ___ ___\
___ | 2*\/ 5 x*\/ 5 |
/ 2 \ 8*\/ 5 *atan|- ------- + -------|
3*log\9 + x - 4*x/ \ 5 5 /
C + ------------------- + ---------------------------------
2 5
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 5 *(-2 + x)| | / 2 \ 8*\/ 5 *atan|--------------| | 3*x + 2 3*log\9 + x - 4*x/ \ 5 / | ------------ dx = C + ------------------- + ---------------------------- | 2 2 5 | x - 4*x + 9 | /
$$\int \frac{3 x + 2}{\left(x^{2} - 4 x\right) + 9}\, dx = C + \frac{3 \log{\left(x^{2} - 4 x + 9 \right)}}{2} + \frac{8 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(x - 2\right)}{5} \right)}}{5}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 5 | ___ |2*\/ 5 |
8*\/ 5 *atan|-----| 8*\/ 5 *atan|-------|
3*log(9) 3*log(6) \ 5 / \ 5 /
- -------- + -------- - ------------------- + ---------------------
2 2 5 5
$$- \frac{3 \log{\left(9 \right)}}{2} - \frac{8 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{5} + \frac{8 \sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)}}{5} + \frac{3 \log{\left(6 \right)}}{2}$$
=
=
/ ___\ / ___\
___ |\/ 5 | ___ |2*\/ 5 |
8*\/ 5 *atan|-----| 8*\/ 5 *atan|-------|
3*log(9) 3*log(6) \ 5 / \ 5 /
- -------- + -------- - ------------------- + ---------------------
2 2 5 5
$$- \frac{3 \log{\left(9 \right)}}{2} - \frac{8 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{5} + \frac{8 \sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5}}{5} \right)}}{5} + \frac{3 \log{\left(6 \right)}}{2}$$
-3*log(9)/2 + 3*log(6)/2 - 8*sqrt(5)*atan(sqrt(5)/5)/5 + 8*sqrt(5)*atan(2*sqrt(5)/5)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.