Sr Examen
Integral de (6x+7)/(3x^2+6x+7) dx
Solución
Ha introducido
[src]
1 / | | 6*x + 7 | -------------- dx | 2 | 3*x + 6*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{6 x + 7}{\left(3 x^{2} + 6 x\right) + 7}\, dx$$
Integral((6*x + 7)/(3*x^2 + 6*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 6*x + 7 | -------------- dx | 2 | 3*x + 6*x + 7 | /
Reescribimos la función subintegral
6*x + 7 3*2*x + 6 1
-------------- = -------------- + ----------------------------
2 2 / 2 \
3*x + 6*x + 7 3*x + 6*x + 7 |/ ___ ___\ |
||-\/ 3 \/ 3 | |
4*||-------*x - -----| + 1|
\\ 2 2 / /o
/ | | 6*x + 7 | -------------- dx | 2 = | 3*x + 6*x + 7 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 2 2 / /
| |
/ | 3*2*x + 6
------------------------------ + | -------------- dx
4 | 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
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 2 2 /
|
/
------------------------------
4 hacemos el cambio
___ ___
\/ 3 x*\/ 3
v = - ----- - -------
2 2 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
4 4 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1 / ___ ___\
| \ 2 2 / ___ |\/ 3 x*\/ 3 |
| \/ 3 *atan|----- + -------|
/ \ 2 2 /
------------------------------ = ---------------------------
4 6 La solución:
/ ___ ___\
___ |\/ 3 x*\/ 3 |
\/ 3 *atan|----- + -------|
\ 2 2 / /7 2 \
C + --------------------------- + log|- + x + 2*x|
6 \3 /
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(1 + x)| | \/ 3 *atan|-------------| | 6*x + 7 \ 2 / / 2 \ | -------------- dx = C + ------------------------- + log\7 + 3*x + 6*x/ | 2 6 | 3*x + 6*x + 7 | /
$$\int \frac{6 x + 7}{\left(3 x^{2} + 6 x\right) + 7}\, dx = C + \log{\left(3 x^{2} + 6 x + 7 \right)} + \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 1\right)}{2} \right)}}{6}$$
Gráfica
Respuesta
[src]
/ ___\
___ |\/ 3 |
\/ 3 *atan|-----| ___
\ 2 / pi*\/ 3
-log(7/3) - ----------------- + -------- + log(16/3)
6 18
$$- \log{\left(\frac{7}{3} \right)} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)}}{6} + \frac{\sqrt{3} \pi}{18} + \log{\left(\frac{16}{3} \right)}$$
=
=
/ ___\
___ |\/ 3 |
\/ 3 *atan|-----| ___
\ 2 / pi*\/ 3
-log(7/3) - ----------------- + -------- + log(16/3)
6 18
$$- \log{\left(\frac{7}{3} \right)} - \frac{\sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)}}{6} + \frac{\sqrt{3} \pi}{18} + \log{\left(\frac{16}{3} \right)}$$
-log(7/3) - sqrt(3)*atan(sqrt(3)/2)/6 + pi*sqrt(3)/18 + log(16/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.