Sr Examen
Integral de (3x+4)/(x^2+4x+7) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 4 | ------------ dx | 2 | x + 4*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 4}{\left(x^{2} + 4 x\right) + 7}\, dx$$
Integral((3*x + 4)/(x^2 + 4*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x + 4 | ------------ dx | 2 | x + 4*x + 7 | /
Reescribimos la función subintegral
2*x + 4
3*------------ /-2 \
2 |---|
3*x + 4 x + 4*x + 7 \ 3 /
------------ = -------------- + --------------------------
2 2 2
x + 4*x + 7 / ___ ___\
|-\/ 3 2*\/ 3 |
|-------*x - -------| + 1
\ 3 3 / o
/ | | 3*x + 4 | ------------ dx | 2 = | x + 4*x + 7 | /
/
|
| 1
2* | -------------------------- dx /
| 2 |
| / ___ ___\ | 2*x + 4
| |-\/ 3 2*\/ 3 | 3* | ------------ dx
| |-------*x - -------| + 1 | 2
| \ 3 3 / | x + 4*x + 7
| |
/ /
- ---------------------------------- + --------------------
3 2 En integral
/
|
| 2*x + 4
3* | ------------ dx
| 2
| x + 4*x + 7
|
/
--------------------
2 hacemos el cambio
2 u = x + 4*x
entonces
integral =
/
|
| 1
3* | ----- du
| 7 + u
|
/ 3*log(7 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 4
3* | ------------ dx
| 2
| x + 4*x + 7
| / 2 \
/ 3*log\7 + x + 4*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
-2* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1
| \ 3 3 /
|
/
-----------------------------------
3 hacemos el cambio
___ ___
2*\/ 3 x*\/ 3
v = - ------- - -------
3 3 entonces
integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
3 3 hacemos cambio inverso
/
|
| 1
-2* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1 / ___ ___\
| \ 3 3 / ___ |2*\/ 3 x*\/ 3 |
| -2*\/ 3 *atan|------- + -------|
/ \ 3 3 /
----------------------------------- = --------------------------------
3 3 La solución:
/ ___ ___\
___ |2*\/ 3 x*\/ 3 |
/ 2 \ 2*\/ 3 *atan|------- + -------|
3*log\7 + x + 4*x/ \ 3 3 /
C + ------------------- - -------------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(2 + x)| | / 2 \ 2*\/ 3 *atan|-------------| | 3*x + 4 3*log\7 + x + 4*x/ \ 3 / | ------------ dx = C + ------------------- - --------------------------- | 2 2 3 | x + 4*x + 7 | /
$$\int \frac{3 x + 4}{\left(x^{2} + 4 x\right) + 7}\, dx = C + \frac{3 \log{\left(x^{2} + 4 x + 7 \right)}}{2} - \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |2*\/ 3 |
___ 2*\/ 3 *atan|-------|
3*log(7) 3*log(12) 2*pi*\/ 3 \ 3 /
- -------- + --------- - ---------- + ---------------------
2 2 9 3
$$- \frac{3 \log{\left(7 \right)}}{2} - \frac{2 \sqrt{3} \pi}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} + \frac{3 \log{\left(12 \right)}}{2}$$
=
=
/ ___\
___ |2*\/ 3 |
___ 2*\/ 3 *atan|-------|
3*log(7) 3*log(12) 2*pi*\/ 3 \ 3 /
- -------- + --------- - ---------- + ---------------------
2 2 9 3
$$- \frac{3 \log{\left(7 \right)}}{2} - \frac{2 \sqrt{3} \pi}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} + \frac{3 \log{\left(12 \right)}}{2}$$
-3*log(7)/2 + 3*log(12)/2 - 2*pi*sqrt(3)/9 + 2*sqrt(3)*atan(2*sqrt(3)/3)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.