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