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