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