Sr Examen
Integral de x/(x^2-2x+8) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | x - 2*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} - 2 x\right) + 8}\, dx$$
Integral(x/(x^2 - 2*x + 8), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 | x - 2*x + 8 | /
Reescribimos la función subintegral
/ 2*x - 2 \
|------------|
| 2 |
x \x - 2*x + 8/ 1
------------ = -------------- + ----------------------------
2 2 / 2 \
x - 2*x + 8 |/ ___ ___\ |
||-\/ 7 \/ 7 | |
7*||-------*x + -----| + 1|
\\ 7 7 / /o
/ | | x | ------------ dx | 2 = | x - 2*x + 8 | /
/
|
| 1
/ | ------------------------ dx
| | 2
| 2*x - 2 | / ___ ___\
| ------------ dx | |-\/ 7 \/ 7 |
| 2 | |-------*x + -----| + 1
| x - 2*x + 8 | \ 7 7 /
| |
/ /
------------------ + ------------------------------
2 7 En integral
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 8
|
/
------------------
2 hacemos el cambio
2 u = x - 2*x
entonces
integral =
/
|
| 1
| ----- du
| 8 + u
|
/ log(8 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x - 2
| ------------ dx
| 2
| x - 2*x + 8
| / 2 \
/ log\8 + x - 2*x/
------------------ = -----------------
2 2 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 7 \/ 7 |
| |-------*x + -----| + 1
| \ 7 7 /
|
/
------------------------------
7 hacemos el cambio
___ ___
\/ 7 x*\/ 7
v = ----- - -------
7 7 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
7 7 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 7 \/ 7 |
| |-------*x + -----| + 1 / ___ ___\
| \ 7 7 / ___ | \/ 7 x*\/ 7 |
| \/ 7 *atan|- ----- + -------|
/ \ 7 7 /
------------------------------ = -----------------------------
7 7 La solución:
/ ___ ___\
___ | \/ 7 x*\/ 7 |
/ 2 \ \/ 7 *atan|- ----- + -------|
log\8 + x - 2*x/ \ 7 7 /
C + ----------------- + -----------------------------
2 7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 7 *(-1 + x)| | / 2 \ \/ 7 *atan|--------------| | x log\8 + x - 2*x/ \ 7 / | ------------ dx = C + ----------------- + -------------------------- | 2 2 7 | x - 2*x + 8 | /
$$\int \frac{x}{\left(x^{2} - 2 x\right) + 8}\, dx = C + \frac{\log{\left(x^{2} - 2 x + 8 \right)}}{2} + \frac{\sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7} \left(x - 1\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\
___ |\/ 7 |
\/ 7 *atan|-----|
log(7) log(8) \ 7 /
------ - ------ + -----------------
2 2 7
$$- \frac{\log{\left(8 \right)}}{2} + \frac{\sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \frac{\log{\left(7 \right)}}{2}$$
=
=
/ ___\
___ |\/ 7 |
\/ 7 *atan|-----|
log(7) log(8) \ 7 /
------ - ------ + -----------------
2 2 7
$$- \frac{\log{\left(8 \right)}}{2} + \frac{\sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \frac{\log{\left(7 \right)}}{2}$$
log(7)/2 - log(8)/2 + sqrt(7)*atan(sqrt(7)/7)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.