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