Sr Examen
Integral de (5x+2)/(4x^2+4x+10) dx
Solución
Ha introducido
[src]
1 / | | 5*x + 2 | --------------- dx | 2 | 4*x + 4*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{5 x + 2}{\left(4 x^{2} + 4 x\right) + 10}\, dx$$
Integral((5*x + 2)/(4*x^2 + 4*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 5*x + 2 | --------------- dx | 2 | 4*x + 4*x + 10 | /
Reescribimos la función subintegral
4*2*x + 4
5*--------------- /-1 \
2 |---|
5*x + 2 4*x + 4*x + 10 \2*9/
--------------- = ----------------- + ----------------
2 8 2
4*x + 4*x + 10 / 2*x 1\
|- --- - -| + 1
\ 3 3/ o
/ | | 5*x + 2 | --------------- dx | 2 = | 4*x + 4*x + 10 | /
/
|
| 1 /
| ---------------- dx |
| 2 | 4*2*x + 4
| / 2*x 1\ 5* | --------------- dx
| |- --- - -| + 1 | 2
| \ 3 3/ | 4*x + 4*x + 10
| |
/ /
- ---------------------- + -----------------------
18 8 En integral
/
|
| 4*2*x + 4
5* | --------------- dx
| 2
| 4*x + 4*x + 10
|
/
-----------------------
8 hacemos el cambio
2 u = 4*x + 4*x
entonces
integral =
/
|
| 1
5* | ------ du
| 10 + u
|
/ 5*log(10 + u)
-------------- = -------------
8 8 hacemos cambio inverso
/
|
| 4*2*x + 4
5* | --------------- dx
| 2
| 4*x + 4*x + 10
| / 2\
/ 5*log\5 + 2*x + 2*x /
----------------------- = ---------------------
8 8 En integral
/
|
| 1
- | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/
|
/
------------------------
18 hacemos el cambio
1 2*x
v = - - - ---
3 3 entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
18 18 hacemos cambio inverso
/
|
| 1
- | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/ /1 2*x\
| -atan|- + ---|
/ \3 3 /
------------------------ = ---------------
18 12 La solución:
/1 2*x\ /5 2\
atan|- + ---| 5*log|- + x + x |
\3 3 / \2 /
C - ------------- + -----------------
12 8
Respuesta (Indefinida)
[src]
/ /1 2*x\ | atan|- + ---| / 2\ | 5*x + 2 \3 3 / 5*log\5 + 2*x + 2*x / | --------------- dx = C - ------------- + --------------------- | 2 12 8 | 4*x + 4*x + 10 | /
$$\int \frac{5 x + 2}{\left(4 x^{2} + 4 x\right) + 10}\, dx = C + \frac{5 \log{\left(2 x^{2} + 2 x + 5 \right)}}{8} - \frac{\operatorname{atan}{\left(\frac{2 x}{3} + \frac{1}{3} \right)}}{12}$$
Gráfica
Respuesta
[src]
5*log(5/2) pi atan(1/3) 5*log(9/2)
- ---------- - -- + --------- + ----------
8 48 12 8
$$- \frac{5 \log{\left(\frac{5}{2} \right)}}{8} - \frac{\pi}{48} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{12} + \frac{5 \log{\left(\frac{9}{2} \right)}}{8}$$
=
=
5*log(5/2) pi atan(1/3) 5*log(9/2)
- ---------- - -- + --------- + ----------
8 48 12 8
$$- \frac{5 \log{\left(\frac{5}{2} \right)}}{8} - \frac{\pi}{48} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{12} + \frac{5 \log{\left(\frac{9}{2} \right)}}{8}$$
-5*log(5/2)/8 - pi/48 + atan(1/3)/12 + 5*log(9/2)/8
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.