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