Sr Examen
Integral de x/(x^2-5x+7) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | x - 5*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 5 x\right) + 7}\, dx$$
Integral(x/(x^2 - 5*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 | x - 5*x + 7 | /
Reescribimos la función subintegral
/ 2*x - 5 \
|------------| / 5 \
| 2 | |-----|
x \x - 5*x + 7/ \2*3/4/
------------ = -------------- + ---------------------------
2 2 2
x - 5*x + 7 / ___ ___\
|-2*\/ 3 5*\/ 3 |
|--------*x + -------| + 1
\ 3 3 / o
/ | | x | ------------ dx | 2 = | x - 5*x + 7 | /
/
|
| 1
/ 10* | --------------------------- dx
| | 2
| 2*x - 5 | / ___ ___\
| ------------ dx | |-2*\/ 3 5*\/ 3 |
| 2 | |--------*x + -------| + 1
| x - 5*x + 7 | \ 3 3 /
| |
/ /
------------------ + ------------------------------------
2 3 En integral
/
|
| 2*x - 5
| ------------ dx
| 2
| x - 5*x + 7
|
/
------------------
2 hacemos el cambio
2 u = x - 5*x
entonces
integral =
/
|
| 1
| ----- du
| 7 + u
|
/ log(7 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x - 5
| ------------ dx
| 2
| x - 5*x + 7
| / 2 \
/ log\7 + x - 5*x/
------------------ = -----------------
2 2 En integral
/
|
| 1
10* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 5*\/ 3 |
| |--------*x + -------| + 1
| \ 3 3 /
|
/
------------------------------------
3 hacemos el cambio
___ ___
5*\/ 3 2*x*\/ 3
v = ------- - ---------
3 3 entonces
integral =
/
|
| 1
10* | ------ dv
| 2
| 1 + v
|
/ 10*atan(v)
--------------- = ----------
3 3 hacemos cambio inverso
/
|
| 1
10* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 5*\/ 3 |
| |--------*x + -------| + 1 / ___ ___\
| \ 3 3 / ___ | 5*\/ 3 2*x*\/ 3 |
| 5*\/ 3 *atan|- ------- + ---------|
/ \ 3 3 /
------------------------------------ = -----------------------------------
3 3 La solución:
/ ___ ___\
___ | 5*\/ 3 2*x*\/ 3 |
/ 2 \ 5*\/ 3 *atan|- ------- + ---------|
log\7 + x - 5*x/ \ 3 3 /
C + ----------------- + -----------------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 3 *(-5/2 + x)| | / 2 \ 5*\/ 3 *atan|------------------| | x log\7 + x - 5*x/ \ 3 / | ------------ dx = C + ----------------- + -------------------------------- | 2 2 3 | x - 5*x + 7 | /
$$\int \frac{x}{\left(x^{2} - 5 x\right) + 7}\, dx = C + \frac{\log{\left(x^{2} - 5 x + 7 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x - \frac{5}{2}\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |5*\/ 3 |
___ 5*\/ 3 *atan|-------|
log(3) log(7) 5*pi*\/ 3 \ 3 /
------ - ------ - ---------- + ---------------------
2 2 9 3
$$- \frac{5 \sqrt{3} \pi}{9} - \frac{\log{\left(7 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3}$$
=
=
/ ___\
___ |5*\/ 3 |
___ 5*\/ 3 *atan|-------|
log(3) log(7) 5*pi*\/ 3 \ 3 /
------ - ------ - ---------- + ---------------------
2 2 9 3
$$- \frac{5 \sqrt{3} \pi}{9} - \frac{\log{\left(7 \right)}}{2} + \frac{\log{\left(3 \right)}}{2} + \frac{5 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3}$$
log(3)/2 - log(7)/2 - 5*pi*sqrt(3)/9 + 5*sqrt(3)*atan(5*sqrt(3)/3)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.