Sr Examen
Integral de dx/(8*x^2-4*x+1)(2*x-1) dx
Solución
Ha introducido
[src]
1 / | | 2*x - 1 | -------------- dx | 2 | 8*x - 4*x + 1 | / 0
$$\int\limits_{0}^{1} \frac{2 x - 1}{\left(8 x^{2} - 4 x\right) + 1}\, dx$$
Integral((2*x - 1)/(8*x^2 - 4*x + 1), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x - 1 | -------------- dx | 2 | 8*x - 4*x + 1 | /
Reescribimos la función subintegral
/ 8*2*x - 4 \
|--------------| / -1 \
| 2 | |-----|
2*x - 1 \8*x - 4*x + 1/ \2*1/2/
-------------- = ---------------- + ---------------
2 8 2
8*x - 4*x + 1 (-4*x + 1) + 1o
/ | | 2*x - 1 | -------------- dx | 2 = | 8*x - 4*x + 1 | /
/
|
| 8*2*x - 4
| -------------- dx
| 2
/ | 8*x - 4*x + 1
| |
| 1 /
- | --------------- dx + --------------------
| 2 8
| (-4*x + 1) + 1
|
/ En integral
/
|
| 8*2*x - 4
| -------------- dx
| 2
| 8*x - 4*x + 1
|
/
--------------------
8 hacemos el cambio
2 u = -4*x + 8*x
entonces
integral =
/
|
| 1
| ----- du
| 1 + u
|
/ log(1 + u)
----------- = ----------
8 8 hacemos cambio inverso
/
|
| 8*2*x - 4
| -------------- dx
| 2
| 8*x - 4*x + 1
| / 2\
/ log\1 - 4*x + 8*x /
-------------------- = -------------------
8 8 En integral
/ | | 1 - | --------------- dx | 2 | (-4*x + 1) + 1 | /
hacemos el cambio
v = 1 - 4*x
entonces
integral =
/ | | 1 - | ------ dv = -atan(v) | 2 | 1 + v | /
hacemos cambio inverso
/ | | 1 -atan(-1 + 4*x) - | --------------- dx = ---------------- | 2 4 | (-4*x + 1) + 1 | /
La solución:
/1 2 x\
log|- + x - -|
atan(-1 + 4*x) \8 2/
C - -------------- + ---------------
4 8
Respuesta (Indefinida)
[src]
/ | / 2\ | 2*x - 1 atan(-1 + 4*x) log\2 - 8*x + 16*x / | -------------- dx = C - -------------- + -------------------- | 2 4 8 | 8*x - 4*x + 1 | /
$$\int \frac{2 x - 1}{\left(8 x^{2} - 4 x\right) + 1}\, dx = C + \frac{\log{\left(16 x^{2} - 8 x + 2 \right)}}{8} - \frac{\operatorname{atan}{\left(4 x - 1 \right)}}{4}$$
Gráfica
Respuesta
[src]
atan(3) pi log(8) log(5/8)
- ------- - -- + ------ + --------
4 16 8 8
$$- \frac{\operatorname{atan}{\left(3 \right)}}{4} - \frac{\pi}{16} + \frac{\log{\left(\frac{5}{8} \right)}}{8} + \frac{\log{\left(8 \right)}}{8}$$
=
=
atan(3) pi log(8) log(5/8)
- ------- - -- + ------ + --------
4 16 8 8
$$- \frac{\operatorname{atan}{\left(3 \right)}}{4} - \frac{\pi}{16} + \frac{\log{\left(\frac{5}{8} \right)}}{8} + \frac{\log{\left(8 \right)}}{8}$$
-atan(3)/4 - pi/16 + log(8)/8 + log(5/8)/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.