Sr Examen
Integral de (2x+1)/(3x^2+6x+30) dx
Solución
Ha introducido
[src]
1 / | | 2*x + 1 | --------------- dx | 2 | 3*x + 6*x + 30 | / 0
$$\int\limits_{0}^{1} \frac{2 x + 1}{\left(3 x^{2} + 6 x\right) + 30}\, dx$$
Integral((2*x + 1)/(3*x^2 + 6*x + 30), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x + 1 | --------------- dx | 2 | 3*x + 6*x + 30 | /
Reescribimos la función subintegral
/ 3*2*x + 6 \
|---------------| /-1 \
| 2 | |---|
2*x + 1 \3*x + 6*x + 30/ \ 27/
--------------- = ----------------- + --------------
2 3 2
3*x + 6*x + 30 / x 1\
|- - - -| + 1
\ 3 3/ o
/ | | 2*x + 1 | --------------- dx | 2 = | 3*x + 6*x + 30 | /
/
|
| 1 /
| -------------- dx |
| 2 | 3*2*x + 6
| / x 1\ | --------------- dx
| |- - - -| + 1 | 2
| \ 3 3/ | 3*x + 6*x + 30
| |
/ /
- -------------------- + ---------------------
27 3 En integral
/
|
| 3*2*x + 6
| --------------- dx
| 2
| 3*x + 6*x + 30
|
/
---------------------
3 hacemos el cambio
2 u = 3*x + 6*x
entonces
integral =
/
|
| 1
| ------ du
| 30 + u
|
/ log(30 + u)
------------ = -----------
3 3 hacemos cambio inverso
/
|
| 3*2*x + 6
| --------------- dx
| 2
| 3*x + 6*x + 30
| / 2 \
/ log\10 + x + 2*x/
--------------------- = ------------------
3 3 En integral
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/
|
/
----------------------
27 hacemos el cambio
1 x
v = - - - -
3 3entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
27 27 hacemos cambio inverso
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/ /1 x\
| -atan|- + -|
/ \3 3/
---------------------- = -------------
27 9 La solución:
/1 x\
atan|- + -| / 2 \
\3 3/ log\10 + x + 2*x/
C - ----------- + ------------------
9 3
Respuesta (Indefinida)
[src]
/ /1 x\ | atan|- + -| / 2 \ | 2*x + 1 \3 3/ log\10 + x + 2*x/ | --------------- dx = C - ----------- + ------------------ | 2 9 3 | 3*x + 6*x + 30 | /
$$\int \frac{2 x + 1}{\left(3 x^{2} + 6 x\right) + 30}\, dx = C + \frac{\log{\left(x^{2} + 2 x + 10 \right)}}{3} - \frac{\operatorname{atan}{\left(\frac{x}{3} + \frac{1}{3} \right)}}{9}$$
Gráfica
Respuesta
[src]
log(10) atan(2/3) log(13) atan(1/3)
- ------- - --------- + ------- + ---------
3 9 3 9
$$- \frac{\log{\left(10 \right)}}{3} - \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{9} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{9} + \frac{\log{\left(13 \right)}}{3}$$
=
=
log(10) atan(2/3) log(13) atan(1/3)
- ------- - --------- + ------- + ---------
3 9 3 9
$$- \frac{\log{\left(10 \right)}}{3} - \frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{9} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{9} + \frac{\log{\left(13 \right)}}{3}$$
-log(10)/3 - atan(2/3)/9 + log(13)/3 + atan(1/3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.