Sr Examen
Integral de (12x+11)dx/(9x^2-6x+2) dx
Solución
Ha introducido
[src]
1 / | | 12*x + 11 | -------------- dx | 2 | 9*x - 6*x + 2 | / 0
$$\int\limits_{0}^{1} \frac{12 x + 11}{\left(9 x^{2} - 6 x\right) + 2}\, dx$$
Integral((12*x + 11)/(9*x^2 - 6*x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 12*x + 11 | -------------- dx | 2 | 9*x - 6*x + 2 | /
Reescribimos la función subintegral
9*2*x - 6
2*-------------- /15\
2 |--|
12*x + 11 9*x - 6*x + 2 \1 /
-------------- = ---------------- + ---------------
2 3 2
9*x - 6*x + 2 (-3*x + 1) + 1o
/ | | 12*x + 11 | -------------- dx | 2 = | 9*x - 6*x + 2 | /
/
|
| 9*2*x - 6
2* | -------------- dx
| 2
/ | 9*x - 6*x + 2
| |
| 1 /
15* | --------------- dx + ----------------------
| 2 3
| (-3*x + 1) + 1
|
/ En integral
/
|
| 9*2*x - 6
2* | -------------- dx
| 2
| 9*x - 6*x + 2
|
/
----------------------
3 hacemos el cambio
2 u = -6*x + 9*x
entonces
integral =
/
|
| 1
2* | ----- du
| 2 + u
|
/ 2*log(2 + u)
------------- = ------------
3 3 hacemos cambio inverso
/
|
| 9*2*x - 6
2* | -------------- dx
| 2
| 9*x - 6*x + 2
| / 2\
/ 2*log\2 - 6*x + 9*x /
---------------------- = ---------------------
3 3 En integral
/
|
| 1
15* | --------------- dx
| 2
| (-3*x + 1) + 1
|
/ hacemos el cambio
v = 1 - 3*x
entonces
integral =
/
|
| 1
15* | ------ dv = 15*atan(v)
| 2
| 1 + v
|
/ hacemos cambio inverso
/
|
| 1
15* | --------------- dx = 5*atan(-1 + 3*x)
| 2
| (-3*x + 1) + 1
|
/ La solución:
/2 2 2*x\
2*log|- + x - ---|
\9 3 /
C + 5*atan(-1 + 3*x) + -------------------
3
Respuesta (Indefinida)
[src]
/ | / 2\ | 12*x + 11 2*log\2 - 6*x + 9*x / | -------------- dx = C + 5*atan(-1 + 3*x) + --------------------- | 2 3 | 9*x - 6*x + 2 | /
$$\int \frac{12 x + 11}{\left(9 x^{2} - 6 x\right) + 2}\, dx = C + \frac{2 \log{\left(9 x^{2} - 6 x + 2 \right)}}{3} + 5 \operatorname{atan}{\left(3 x - 1 \right)}$$
Gráfica
Respuesta
[src]
2*log(2/9) 2*log(5/9) 5*pi
5*atan(2) - ---------- + ---------- + ----
3 3 4
$$\frac{2 \log{\left(\frac{5}{9} \right)}}{3} - \frac{2 \log{\left(\frac{2}{9} \right)}}{3} + \frac{5 \pi}{4} + 5 \operatorname{atan}{\left(2 \right)}$$
=
=
2*log(2/9) 2*log(5/9) 5*pi
5*atan(2) - ---------- + ---------- + ----
3 3 4
$$\frac{2 \log{\left(\frac{5}{9} \right)}}{3} - \frac{2 \log{\left(\frac{2}{9} \right)}}{3} + \frac{5 \pi}{4} + 5 \operatorname{atan}{\left(2 \right)}$$
5*atan(2) - 2*log(2/9)/3 + 2*log(5/9)/3 + 5*pi/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.