Sr Examen
Integral de (5x+2)/(xx+2x+10) dx
Solución
Ha introducido
[src]
1 / | | 5*x + 2 | -------------- dx | x*x + 2*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{5 x + 2}{\left(x x + 2 x\right) + 10}\, dx$$
Integral((5*x + 2)/(x*x + 2*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 5*x + 2 | -------------- dx | x*x + 2*x + 10 | /
Reescribimos la función subintegral
2*x + 2
5*------------- /-3 \
2 |---|
5*x + 2 x + 2*x + 10 \ 9 /
-------------- = --------------- + --------------
x*x + 2*x + 10 2 2
/ x 1\
|- - - -| + 1
\ 3 3/ o
/ | | 5*x + 2 | -------------- dx = | x*x + 2*x + 10 | /
/
|
| 1 /
| -------------- dx |
| 2 | 2*x + 2
| / x 1\ 5* | ------------- dx
| |- - - -| + 1 | 2
| \ 3 3/ | x + 2*x + 10
| |
/ /
- -------------------- + ---------------------
3 2 En integral
/
|
| 2*x + 2
5* | ------------- dx
| 2
| x + 2*x + 10
|
/
---------------------
2 hacemos el cambio
2 u = x + 2*x
entonces
integral =
/
|
| 1
5* | ------ du
| 10 + u
|
/ 5*log(10 + u)
-------------- = -------------
2 2 hacemos cambio inverso
/
|
| 2*x + 2
5* | ------------- dx
| 2
| x + 2*x + 10
| / 2 \
/ 5*log\10 + x + 2*x/
--------------------- = --------------------
2 2 En integral
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/
|
/
----------------------
3 hacemos el cambio
1 x
v = - - - -
3 3entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
3 3 hacemos cambio inverso
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 3 3/
|
/ /1 x\
---------------------- = -atan|- + -|
3 \3 3/La solución:
/ 2 \
/1 x\ 5*log\10 + x + 2*x/
C - atan|- + -| + --------------------
\3 3/ 2
Respuesta (Indefinida)
[src]
/ | / 2 \ | 5*x + 2 /1 x\ 5*log\10 + x + 2*x/ | -------------- dx = C - atan|- + -| + -------------------- | x*x + 2*x + 10 \3 3/ 2 | /
$$\int \frac{5 x + 2}{\left(x x + 2 x\right) + 10}\, dx = C + \frac{5 \log{\left(x^{2} + 2 x + 10 \right)}}{2} - \operatorname{atan}{\left(\frac{x}{3} + \frac{1}{3} \right)}$$
Gráfica
Respuesta
[src]
5*log(10) 5*log(13)
-atan(2/3) - --------- + --------- + atan(1/3)
2 2
$$- \frac{5 \log{\left(10 \right)}}{2} - \operatorname{atan}{\left(\frac{2}{3} \right)} + \operatorname{atan}{\left(\frac{1}{3} \right)} + \frac{5 \log{\left(13 \right)}}{2}$$
=
=
5*log(10) 5*log(13)
-atan(2/3) - --------- + --------- + atan(1/3)
2 2
$$- \frac{5 \log{\left(10 \right)}}{2} - \operatorname{atan}{\left(\frac{2}{3} \right)} + \operatorname{atan}{\left(\frac{1}{3} \right)} + \frac{5 \log{\left(13 \right)}}{2}$$
-atan(2/3) - 5*log(10)/2 + 5*log(13)/2 + atan(1/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.