Sr Examen
Integral de (3-4*x)/((x^2)+4*x+7) dx
Solución
Ha introducido
[src]
1 / | | 3 - 4*x | ------------ dx | 2 | x + 4*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{3 - 4 x}{\left(x^{2} + 4 x\right) + 7}\, dx$$
Integral((3 - 4*x)/(x^2 + 4*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3 - 4*x | ------------ dx | 2 | x + 4*x + 7 | /
Reescribimos la función subintegral
/11\
|--|
3 - 4*x 2*x + 4 \3 /
------------ = - 2*------------ + --------------------------
2 2 2
x + 4*x + 7 x + 4*x + 7 / ___ ___\
|-\/ 3 2*\/ 3 |
|-------*x - -------| + 1
\ 3 3 / o
/ | | 3 - 4*x | ------------ dx | 2 = | x + 4*x + 7 | /
/
|
| 1
11* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1
/ | \ 3 3 /
| |
| 2*x + 4 /
- 2* | ------------ dx + -----------------------------------
| 2 3
| x + 4*x + 7
|
/ En integral
/
|
| 2*x + 4
-2* | ------------ dx
| 2
| x + 4*x + 7
|
/ hacemos el cambio
2 u = x + 4*x
entonces
integral =
/
|
| 1
-2* | ----- du = -2*log(7 + u)
| 7 + u
|
/ hacemos cambio inverso
/
|
| 2*x + 4 / 2 \
-2* | ------------ dx = -2*log\7 + x + 4*x/
| 2
| x + 4*x + 7
|
/ En integral
/
|
| 1
11* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1
| \ 3 3 /
|
/
-----------------------------------
3 hacemos el cambio
___ ___
2*\/ 3 x*\/ 3
v = - ------- - -------
3 3 entonces
integral =
/
|
| 1
11* | ------ dv
| 2
| 1 + v
|
/ 11*atan(v)
--------------- = ----------
3 3 hacemos cambio inverso
/
|
| 1
11* | -------------------------- dx
| 2
| / ___ ___\
| |-\/ 3 2*\/ 3 |
| |-------*x - -------| + 1 / ___ ___\
| \ 3 3 / ___ |2*\/ 3 x*\/ 3 |
| 11*\/ 3 *atan|------- + -------|
/ \ 3 3 /
----------------------------------- = --------------------------------
3 3 La solución:
/ ___ ___\
___ |2*\/ 3 x*\/ 3 |
11*\/ 3 *atan|------- + -------|
/ 2 \ \ 3 3 /
C - 2*log\7 + x + 4*x/ + --------------------------------
3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(2 + x)| | 11*\/ 3 *atan|-------------| | 3 - 4*x / 2 \ \ 3 / | ------------ dx = C - 2*log\7 + x + 4*x/ + ---------------------------- | 2 3 | x + 4*x + 7 | /
$$\int \frac{3 - 4 x}{\left(x^{2} + 4 x\right) + 7}\, dx = C - 2 \log{\left(x^{2} + 4 x + 7 \right)} + \frac{11 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 2\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |2*\/ 3 |
11*\/ 3 *atan|-------| ___
\ 3 / 11*pi*\/ 3
-2*log(12) + 2*log(7) - ---------------------- + -----------
3 9
$$- \frac{11 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} - 2 \log{\left(12 \right)} + 2 \log{\left(7 \right)} + \frac{11 \sqrt{3} \pi}{9}$$
=
=
/ ___\
___ |2*\/ 3 |
11*\/ 3 *atan|-------| ___
\ 3 / 11*pi*\/ 3
-2*log(12) + 2*log(7) - ---------------------- + -----------
3 9
$$- \frac{11 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3} - 2 \log{\left(12 \right)} + 2 \log{\left(7 \right)} + \frac{11 \sqrt{3} \pi}{9}$$
-2*log(12) + 2*log(7) - 11*sqrt(3)*atan(2*sqrt(3)/3)/3 + 11*pi*sqrt(3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.