Sr Examen
Integral de (3x+5)/(3+2x+x^2) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 5 | ------------ dx | 2 | 3 + 2*x + x | / 0
$$\int\limits_{0}^{1} \frac{3 x + 5}{x^{2} + \left(2 x + 3\right)}\, dx$$
Integral((3*x + 5)/(3 + 2*x + x^2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x + 5 | ------------ dx | 2 | 3 + 2*x + x | /
Reescribimos la función subintegral
2*x + 2
3*------------ /2\
2 |-|
3*x + 5 x + 2*x + 3 \2/
------------ = -------------- + ------------------------
2 2 2
3 + 2*x + x / ___ ___\
|-\/ 2 \/ 2 |
|-------*x - -----| + 1
\ 2 2 / o
/ | | 3*x + 5 | ------------ dx | 2 = | 3 + 2*x + x | /
/
|
| 2*x + 2
3* | ------------ dx
| 2
| x + 2*x + 3 /
| |
/ | 1
-------------------- + | ------------------------ dx
2 | 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x - -----| + 1
| \ 2 2 /
|
/ En integral
/
|
| 2*x + 2
3* | ------------ dx
| 2
| x + 2*x + 3
|
/
--------------------
2 hacemos el cambio
2 u = x + 2*x
entonces
integral =
/
|
| 1
3* | ----- du
| 3 + u
|
/ 3*log(3 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 2
3* | ------------ dx
| 2
| x + 2*x + 3
| / 2 \
/ 3*log\3 + x + 2*x/
-------------------- = -------------------
2 2 En integral
/ | | 1 | ------------------------ dx | 2 | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 | \ 2 2 / | /
hacemos el cambio
___ ___
\/ 2 x*\/ 2
v = - ----- - -------
2 2 entonces
integral =
/ | | 1 | ------ dv = atan(v) | 2 | 1 + v | /
hacemos cambio inverso
/ | / ___ ___\ | 1 ___ |\/ 2 x*\/ 2 | | ------------------------ dx = \/ 2 *atan|----- + -------| | 2 \ 2 2 / | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 | \ 2 2 / | /
La solución:
/ 2 \ / ___ ___\
3*log\3 + x + 2*x/ ___ |\/ 2 x*\/ 2 |
C + ------------------- + \/ 2 *atan|----- + -------|
2 \ 2 2 /
Respuesta (Indefinida)
[src]
/ | / 2 \ / ___ \ | 3*x + 5 3*log\3 + x + 2*x/ ___ |\/ 2 *(1 + x)| | ------------ dx = C + ------------------- + \/ 2 *atan|-------------| | 2 2 \ 2 / | 3 + 2*x + x | /
$$\int \frac{3 x + 5}{x^{2} + \left(2 x + 3\right)}\, dx = C + \frac{3 \log{\left(x^{2} + 2 x + 3 \right)}}{2} + \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}$$
Gráfica
Respuesta
[src]
/ ___\
3*log(3) 3*log(6) ___ / ___\ ___ |\/ 2 |
- -------- + -------- + \/ 2 *atan\\/ 2 / - \/ 2 *atan|-----|
2 2 \ 2 /
$$- \frac{3 \log{\left(3 \right)}}{2} - \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} + \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} + \frac{3 \log{\left(6 \right)}}{2}$$
=
=
/ ___\
3*log(3) 3*log(6) ___ / ___\ ___ |\/ 2 |
- -------- + -------- + \/ 2 *atan\\/ 2 / - \/ 2 *atan|-----|
2 2 \ 2 /
$$- \frac{3 \log{\left(3 \right)}}{2} - \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} + \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} + \frac{3 \log{\left(6 \right)}}{2}$$
-3*log(3)/2 + 3*log(6)/2 + sqrt(2)*atan(sqrt(2)) - sqrt(2)*atan(sqrt(2)/2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.