Sr Examen
Integral de (6x)/(3x^2+6x+7) dx
Solución
Ha introducido
[src]
1 / | | 6*x | -------------- dx | 2 | 3*x + 6*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{6 x}{\left(3 x^{2} + 6 x\right) + 7}\, dx$$
Integral((6*x)/(3*x^2 + 6*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 6*x | -------------- dx | 2 | 3*x + 6*x + 7 | /
Reescribimos la función subintegral
/-6 \
|---|
6*x 3*2*x + 6 \ 4 /
-------------- = -------------- + ------------------------
2 2 2
3*x + 6*x + 7 3*x + 6*x + 7 / ___ ___\
|-\/ 3 \/ 3 |
|-------*x - -----| + 1
\ 2 2 / o
/ | | 6*x | -------------- dx | 2 = | 3*x + 6*x + 7 | /
/
|
| 1
3* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 2 2 / /
| |
/ | 3*2*x + 6
- -------------------------------- + | -------------- dx
2 | 2
| 3*x + 6*x + 7
|
/ En integral
/ | | 3*2*x + 6 | -------------- dx | 2 | 3*x + 6*x + 7 | /
hacemos el cambio
2 u = 3*x + 6*x
entonces
integral =
/ | | 1 | ----- du = log(7 + u) | 7 + u | /
hacemos cambio inverso
/ | | 3*2*x + 6 / 2 \ | -------------- dx = log\7 + 3*x + 6*x/ | 2 | 3*x + 6*x + 7 | /
En integral
/
|
| 1
-3* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 2 2 /
|
/
---------------------------------
2 hacemos el cambio
___ ___
\/ 3 x*\/ 3
v = - ----- - -------
2 2 entonces
integral =
/
|
| 1
-3* | ------ dv
| 2
| 1 + v
|
/ -3*atan(v)
--------------- = ----------
2 2 hacemos cambio inverso
/
|
| 1
-3* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 2 2 /
| / ___ ___\
/ ___ |\/ 3 x*\/ 3 |
--------------------------------- = -\/ 3 *atan|----- + -------|
2 \ 2 2 /La solución:
/ ___ ___\
___ |\/ 3 x*\/ 3 | /7 2 \
C - \/ 3 *atan|----- + -------| + log|- + x + 2*x|
\ 2 2 / \3 /
Respuesta (Indefinida)
[src]
/ | / ___ \ | 6*x ___ |\/ 3 *(1 + x)| / 2 \ | -------------- dx = C - \/ 3 *atan|-------------| + log\7 + 3*x + 6*x/ | 2 \ 2 / | 3*x + 6*x + 7 | /
$$\int \frac{6 x}{\left(3 x^{2} + 6 x\right) + 7}\, dx = C + \log{\left(3 x^{2} + 6 x + 7 \right)} - \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 1\right)}{2} \right)}$$
Gráfica
Respuesta
[src]
/ ___\ ___
___ |\/ 3 | pi*\/ 3
-log(7/3) + \/ 3 *atan|-----| - -------- + log(16/3)
\ 2 / 3
$$- \frac{\sqrt{3} \pi}{3} - \log{\left(\frac{7}{3} \right)} + \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)} + \log{\left(\frac{16}{3} \right)}$$
=
=
/ ___\ ___
___ |\/ 3 | pi*\/ 3
-log(7/3) + \/ 3 *atan|-----| - -------- + log(16/3)
\ 2 / 3
$$- \frac{\sqrt{3} \pi}{3} - \log{\left(\frac{7}{3} \right)} + \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)} + \log{\left(\frac{16}{3} \right)}$$
-log(7/3) + sqrt(3)*atan(sqrt(3)/2) - pi*sqrt(3)/3 + log(16/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.