Sr Examen
Integral de x/(x^2-4x+7) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | x - 4*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 4 x\right) + 7}\, dx$$
Integral(x/(x^2 - 4*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 | x - 4*x + 7 | /
Reescribimos la función subintegral
/ 2*x - 4 \
|------------| /2\
| 2 | |-|
x \x - 4*x + 7/ \3/
------------ = -------------- + --------------------------
2 2 2
x - 4*x + 7 / ___ ___\
|-\/ 3 2*\/ 3 |
|-------*x + -------| + 1
\ 3 3 / o
/ | | x | ------------ dx | 2 = | x - 4*x + 7 | /
/
|
| 1
/ 2* | -------------------------- dx
| | 2
| 2*x - 4 | / ___ ___\
| ------------ dx | |-\/ 3 2*\/ 3 |
| 2 | |-------*x + -------| + 1
| x - 4*x + 7 | \ 3 3 /
| |
/ /
------------------ + ----------------------------------
2 3 En integral
/
|
| 2*x - 4
| ------------ dx
| 2
| x - 4*x + 7
|
/
------------------
2 hacemos el cambio
2 u = x - 4*x
entonces
integral =
/
|
| 1
| ----- du
| 7 + u
|
/ log(7 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x - 4
| ------------ dx
| 2
| x - 4*x + 7
| / 2 \
/ 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|- ------- + -------|
log\7 + x - 4*x/ \ 3 3 /
C + ----------------- + ---------------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(-2 + x)| | / 2 \ 2*\/ 3 *atan|--------------| | x log\7 + x - 4*x/ \ 3 / | ------------ dx = C + ----------------- + ---------------------------- | 2 2 3 | x - 4*x + 7 | /
$$\int \frac{x}{\left(x^{2} - 4 x\right) + 7}\, dx = C + \frac{\log{\left(x^{2} - 4 x + 7 \right)}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x - 2\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |2*\/ 3 |
___ 2*\/ 3 *atan|-------|
log(4) log(7) pi*\/ 3 \ 3 /
------ - ------ - -------- + ---------------------
2 2 9 3
$$- \frac{\log{\left(7 \right)}}{2} - \frac{\sqrt{3} \pi}{9} + \frac{\log{\left(4 \right)}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3}$$
=
=
/ ___\
___ |2*\/ 3 |
___ 2*\/ 3 *atan|-------|
log(4) log(7) pi*\/ 3 \ 3 /
------ - ------ - -------- + ---------------------
2 2 9 3
$$- \frac{\log{\left(7 \right)}}{2} - \frac{\sqrt{3} \pi}{9} + \frac{\log{\left(4 \right)}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3}}{3} \right)}}{3}$$
log(4)/2 - log(7)/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.