Sr Examen
Integral de (3x+2)/(4x²+4x+5) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 2 | -------------- dx | 2 | 4*x + 4*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 2}{\left(4 x^{2} + 4 x\right) + 5}\, dx$$
Integral((3*x + 2)/(4*x^2 + 4*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x + 2 | -------------- dx | 2 | 4*x + 4*x + 5 | /
Reescribimos la función subintegral
4*2*x + 4
3*-------------- / 1 \
2 |---|
3*x + 2 4*x + 4*x + 5 \2*4/
-------------- = ---------------- + ---------------
2 8 2
4*x + 4*x + 5 (-x - 1/2) + 1o
/ | | 3*x + 2 | -------------- dx | 2 = | 4*x + 4*x + 5 | /
/ /
| |
| 1 | 4*2*x + 4
| --------------- dx 3* | -------------- dx
| 2 | 2
| (-x - 1/2) + 1 | 4*x + 4*x + 5
| |
/ /
--------------------- + ----------------------
8 8 En integral
/
|
| 4*2*x + 4
3* | -------------- dx
| 2
| 4*x + 4*x + 5
|
/
----------------------
8 hacemos el cambio
2 u = 4*x + 4*x
entonces
integral =
/
|
| 1
3* | ----- du
| 5 + u
|
/ 3*log(5 + u)
------------- = ------------
8 8 hacemos cambio inverso
/
|
| 4*2*x + 4
3* | -------------- dx
| 2
| 4*x + 4*x + 5
| / 2\
/ 3*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\
3*log|- + x + x |
atan(1/2 + x) \4 /
C + ------------- + -----------------
8 8
Respuesta (Indefinida)
[src]
/ | / 2\ | 3*x + 2 atan(1/2 + x) 3*log\5 + 4*x + 4*x / | -------------- dx = C + ------------- + --------------------- | 2 8 8 | 4*x + 4*x + 5 | /
$$\int \frac{3 x + 2}{\left(4 x^{2} + 4 x\right) + 5}\, dx = C + \frac{3 \log{\left(4 x^{2} + 4 x + 5 \right)}}{8} + \frac{\operatorname{atan}{\left(x + \frac{1}{2} \right)}}{8}$$
Gráfica
Respuesta
[src]
3*log(5/4) atan(1/2) atan(3/2) 3*log(13/4)
- ---------- - --------- + --------- + -----------
8 8 8 8
$$- \frac{3 \log{\left(\frac{5}{4} \right)}}{8} - \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{8} + \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{8} + \frac{3 \log{\left(\frac{13}{4} \right)}}{8}$$
=
=
3*log(5/4) atan(1/2) atan(3/2) 3*log(13/4)
- ---------- - --------- + --------- + -----------
8 8 8 8
$$- \frac{3 \log{\left(\frac{5}{4} \right)}}{8} - \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{8} + \frac{\operatorname{atan}{\left(\frac{3}{2} \right)}}{8} + \frac{3 \log{\left(\frac{13}{4} \right)}}{8}$$
-3*log(5/4)/8 - atan(1/2)/8 + atan(3/2)/8 + 3*log(13/4)/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.