Sr Examen
Integral de (4x-3)/(5x^2+6x+18) dx
Solución
Ha introducido
[src]
1 / | | 4*x - 3 | --------------- dx | 2 | 5*x + 6*x + 18 | / 0
$$\int\limits_{0}^{1} \frac{4 x - 3}{\left(5 x^{2} + 6 x\right) + 18}\, dx$$
Integral((4*x - 3)/(5*x^2 + 6*x + 18), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 4*x - 3 | --------------- dx | 2 | 5*x + 6*x + 18 | /
Reescribimos la función subintegral
5*2*x + 6
2*--------------- / -27 \
2 |------|
4*x - 3 5*x + 6*x + 18 \5*81/5/
--------------- = ----------------- + ----------------
2 5 2
5*x + 6*x + 18 / 5*x 1\
|- --- - -| + 1
\ 9 3/ o
/ | | 4*x - 3 | --------------- dx | 2 = | 5*x + 6*x + 18 | /
/
|
| 1 /
| ---------------- dx |
| 2 | 5*2*x + 6
| / 5*x 1\ 2* | --------------- dx
| |- --- - -| + 1 | 2
| \ 9 3/ | 5*x + 6*x + 18
| |
/ /
- ---------------------- + -----------------------
3 5 En integral
/
|
| 5*2*x + 6
2* | --------------- dx
| 2
| 5*x + 6*x + 18
|
/
-----------------------
5 hacemos el cambio
2 u = 5*x + 6*x
entonces
integral =
/
|
| 1
2* | ------ du
| 18 + u
|
/ 2*log(18 + u)
-------------- = -------------
5 5 hacemos cambio inverso
/
|
| 5*2*x + 6
2* | --------------- dx
| 2
| 5*x + 6*x + 18
| / 2 \
/ 2*log\18 + 5*x + 6*x/
----------------------- = ----------------------
5 5 En integral
/
|
| 1
- | ---------------- dx
| 2
| / 5*x 1\
| |- --- - -| + 1
| \ 9 3/
|
/
------------------------
3 hacemos el cambio
1 5*x
v = - - - ---
3 9 entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
3 3 hacemos cambio inverso
/
|
| 1
- | ---------------- dx
| 2
| / 5*x 1\
| |- --- - -| + 1
| \ 9 3/ /1 5*x\
| -3*atan|- + ---|
/ \3 9 /
------------------------ = ----------------
3 5 La solución:
/1 5*x\ /18 2 6*x\
3*atan|- + ---| 2*log|-- + x + ---|
\3 9 / \5 5 /
C - --------------- + --------------------
5 5
Respuesta (Indefinida)
[src]
/ /1 5*x\ | 3*atan|- + ---| / 2 \ | 4*x - 3 \3 9 / 2*log\18 + 5*x + 6*x/ | --------------- dx = C - --------------- + ---------------------- | 2 5 5 | 5*x + 6*x + 18 | /
$$\int \frac{4 x - 3}{\left(5 x^{2} + 6 x\right) + 18}\, dx = C + \frac{2 \log{\left(5 x^{2} + 6 x + 18 \right)}}{5} - \frac{3 \operatorname{atan}{\left(\frac{5 x}{9} + \frac{1}{3} \right)}}{5}$$
Gráfica
Respuesta
[src]
3*atan(8/9) 2*log(18/5) 2*log(29/5) 3*atan(1/3)
- ----------- - ----------- + ----------- + -----------
5 5 5 5
$$- \frac{2 \log{\left(\frac{18}{5} \right)}}{5} - \frac{3 \operatorname{atan}{\left(\frac{8}{9} \right)}}{5} + \frac{3 \operatorname{atan}{\left(\frac{1}{3} \right)}}{5} + \frac{2 \log{\left(\frac{29}{5} \right)}}{5}$$
=
=
3*atan(8/9) 2*log(18/5) 2*log(29/5) 3*atan(1/3)
- ----------- - ----------- + ----------- + -----------
5 5 5 5
$$- \frac{2 \log{\left(\frac{18}{5} \right)}}{5} - \frac{3 \operatorname{atan}{\left(\frac{8}{9} \right)}}{5} + \frac{3 \operatorname{atan}{\left(\frac{1}{3} \right)}}{5} + \frac{2 \log{\left(\frac{29}{5} \right)}}{5}$$
-3*atan(8/9)/5 - 2*log(18/5)/5 + 2*log(29/5)/5 + 3*atan(1/3)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.