Sr Examen
Integral de (x+1)/(4x^2-12x+13) dx
Solución
Ha introducido
[src]
1 / | | x + 1 | ---------------- dx | 2 | 4*x - 12*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{x + 1}{\left(4 x^{2} - 12 x\right) + 13}\, dx$$
Integral((x + 1)/(4*x^2 - 12*x + 13), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 1 | ---------------- dx | 2 | 4*x - 12*x + 13 | /
Reescribimos la función subintegral
/ 4*2*x - 12 \
|----------------| / 5 \
| 2 | |---|
x + 1 \4*x - 12*x + 13/ \2*4/
---------------- = ------------------ + ---------------
2 8 2
4*x - 12*x + 13 (-x + 3/2) + 1o
/ | | x + 1 | ---------------- dx | 2 = | 4*x - 12*x + 13 | /
/ /
| |
| 4*2*x - 12 | 1
| ---------------- dx 5* | --------------- dx
| 2 | 2
| 4*x - 12*x + 13 | (-x + 3/2) + 1
| |
/ /
---------------------- + -----------------------
8 8 En integral
/
|
| 4*2*x - 12
| ---------------- dx
| 2
| 4*x - 12*x + 13
|
/
----------------------
8 hacemos el cambio
2 u = -12*x + 4*x
entonces
integral =
/
|
| 1
| ------ du
| 13 + u
|
/ log(13 + u)
------------ = -----------
8 8 hacemos cambio inverso
/
|
| 4*2*x - 12
| ---------------- dx
| 2
| 4*x - 12*x + 13
| / 2\
/ log\13 - 12*x + 4*x /
---------------------- = ---------------------
8 8 En integral
/
|
| 1
5* | --------------- dx
| 2
| (-x + 3/2) + 1
|
/
-----------------------
8 hacemos el cambio
v = 3/2 - x
entonces
integral =
/
|
| 1
5* | ------ dv
| 2
| 1 + v
|
/ 5*atan(v)
-------------- = ---------
8 8 hacemos cambio inverso
/
|
| 1
5* | --------------- dx
| 2
| (-x + 3/2) + 1
|
/ 5*atan(-3/2 + x)
----------------------- = ----------------
8 8 La solución:
/13 2 \
log|-- + x - 3*x|
\4 / 5*atan(-3/2 + x)
C + ------------------ + ----------------
8 8
Respuesta (Indefinida)
[src]
/ | / 2\ | x + 1 log\13 - 12*x + 4*x / 5*atan(-3/2 + x) | ---------------- dx = C + --------------------- + ---------------- | 2 8 8 | 4*x - 12*x + 13 | /
$$\int \frac{x + 1}{\left(4 x^{2} - 12 x\right) + 13}\, dx = C + \frac{\log{\left(4 x^{2} - 12 x + 13 \right)}}{8} + \frac{5 \operatorname{atan}{\left(x - \frac{3}{2} \right)}}{8}$$
Gráfica
Respuesta
[src]
5*atan(1/2) log(13/4) log(5/4) 5*atan(3/2)
- ----------- - --------- + -------- + -----------
8 8 8 8
$$- \frac{5 \operatorname{atan}{\left(\frac{1}{2} \right)}}{8} - \frac{\log{\left(\frac{13}{4} \right)}}{8} + \frac{\log{\left(\frac{5}{4} \right)}}{8} + \frac{5 \operatorname{atan}{\left(\frac{3}{2} \right)}}{8}$$
=
=
5*atan(1/2) log(13/4) log(5/4) 5*atan(3/2)
- ----------- - --------- + -------- + -----------
8 8 8 8
$$- \frac{5 \operatorname{atan}{\left(\frac{1}{2} \right)}}{8} - \frac{\log{\left(\frac{13}{4} \right)}}{8} + \frac{\log{\left(\frac{5}{4} \right)}}{8} + \frac{5 \operatorname{atan}{\left(\frac{3}{2} \right)}}{8}$$
-5*atan(1/2)/8 - log(13/4)/8 + log(5/4)/8 + 5*atan(3/2)/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.