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