Sr Examen
Integral de (3-x)/(x^2+2x+3) dx
Solución
Ha introducido
[src]
1 / | | 3 - x | ------------ dx | 2 | x + 2*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{3 - x}{\left(x^{2} + 2 x\right) + 3}\, dx$$
Integral((3 - x)/(x^2 + 2*x + 3), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3 - x | ------------ dx | 2 | x + 2*x + 3 | /
Reescribimos la función subintegral
/ 2*x + 2 \
|------------| /4\
| 2 | |-|
3 - x \x + 2*x + 3/ \2/
------------ = - -------------- + ------------------------
2 2 2
x + 2*x + 3 / ___ ___\
|-\/ 2 \/ 2 |
|-------*x - -----| + 1
\ 2 2 / o
/ | | 3 - x | ------------ dx | 2 = | x + 2*x + 3 | /
/
|
| 2*x + 2
| ------------ dx
| 2
/ | x + 2*x + 3
| |
| 1 /
2* | ------------------------ dx - ------------------
| 2 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x - -----| + 1
| \ 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 2* | ------------------------ dx | 2 | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 | \ 2 2 / | /
hacemos el cambio
___ ___
\/ 2 x*\/ 2
v = - ----- - -------
2 2 entonces
integral =
/ | | 1 2* | ------ dv = 2*atan(v) | 2 | 1 + v | /
hacemos cambio inverso
/ | / ___ ___\ | 1 ___ |\/ 2 x*\/ 2 | 2* | ------------------------ dx = 2*\/ 2 *atan|----- + -------| | 2 \ 2 2 / | / ___ ___\ | |-\/ 2 \/ 2 | | |-------*x - -----| + 1 | \ 2 2 / | /
La solución:
/ 2 \ / ___ ___\
log\3 + x + 2*x/ ___ |\/ 2 x*\/ 2 |
C - ----------------- + 2*\/ 2 *atan|----- + -------|
2 \ 2 2 /
Respuesta (Indefinida)
[src]
/ | / 2 \ / ___ \ | 3 - x log\3 + x + 2*x/ ___ |\/ 2 *(1 + x)| | ------------ dx = C - ----------------- + 2*\/ 2 *atan|-------------| | 2 2 \ 2 / | x + 2*x + 3 | /
$$\int \frac{3 - x}{\left(x^{2} + 2 x\right) + 3}\, dx = C - \frac{\log{\left(x^{2} + 2 x + 3 \right)}}{2} + 2 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 1\right)}{2} \right)}$$
Gráfica
Respuesta
[src]
/ ___\ log(3) log(6) ___ |\/ 2 | ___ / ___\ ------ - ------ - 2*\/ 2 *atan|-----| + 2*\/ 2 *atan\\/ 2 / 2 2 \ 2 /
$$- 2 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} - \frac{\log{\left(6 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + 2 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}$$
=
=
/ ___\ log(3) log(6) ___ |\/ 2 | ___ / ___\ ------ - ------ - 2*\/ 2 *atan|-----| + 2*\/ 2 *atan\\/ 2 / 2 2 \ 2 /
$$- 2 \sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)} - \frac{\log{\left(6 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + 2 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)}$$
log(3)/2 - log(6)/2 - 2*sqrt(2)*atan(sqrt(2)/2) + 2*sqrt(2)*atan(sqrt(2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.