Sr Examen
Integral de (x-1)/(x^2+7x+13) dx
Solución
Ha introducido
[src]
1 / | | x - 1 | ------------- dx | 2 | x + 7*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{x - 1}{\left(x^{2} + 7 x\right) + 13}\, dx$$
Integral((x - 1)/(x^2 + 7*x + 13), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x - 1 | ------------- dx | 2 | x + 7*x + 13 | /
Reescribimos la función subintegral
/ 2*x + 7 \
|-------------| / -9 \
| 2 | |-----|
x - 1 \x + 7*x + 13/ \2*3/4/
------------- = --------------- + ---------------------------
2 2 2
x + 7*x + 13 / ___ ___\
|-2*\/ 3 7*\/ 3 |
|--------*x - -------| + 1
\ 3 3 / o
/ | | x - 1 | ------------- dx | 2 = | x + 7*x + 13 | /
/
|
| 2*x + 7
| ------------- dx
| 2
| x + 7*x + 13 /
| |
/ | 1
------------------- - 6* | --------------------------- dx
2 | 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1
| \ 3 3 /
|
/ En integral
/
|
| 2*x + 7
| ------------- dx
| 2
| x + 7*x + 13
|
/
-------------------
2 hacemos el cambio
2 u = x + 7*x
entonces
integral =
/
|
| 1
| ------ du
| 13 + u
|
/ log(13 + u)
------------ = -----------
2 2 hacemos cambio inverso
/
|
| 2*x + 7
| ------------- dx
| 2
| x + 7*x + 13
| / 2 \
/ log\13 + x + 7*x/
------------------- = ------------------
2 2 En integral
/
|
| 1
-6* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1
| \ 3 3 /
|
/ hacemos el cambio
___ ___
7*\/ 3 2*x*\/ 3
v = - ------- - ---------
3 3 entonces
integral =
/
|
| 1
-6* | ------ dv = -6*atan(v)
| 2
| 1 + v
|
/ hacemos cambio inverso
/
| / ___ ___\
| 1 ___ |7*\/ 3 2*x*\/ 3 |
-6* | --------------------------- dx = -3*\/ 3 *atan|------- + ---------|
| 2 \ 3 3 /
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1
| \ 3 3 /
|
/ La solución:
/ 2 \ / ___ ___\
log\13 + x + 7*x/ ___ |7*\/ 3 2*x*\/ 3 |
C + ------------------ - 3*\/ 3 *atan|------- + ---------|
2 \ 3 3 /
Respuesta (Indefinida)
[src]
/ | / 2 \ / ___ \ | x - 1 log\13 + x + 7*x/ ___ |2*\/ 3 *(7/2 + x)| | ------------- dx = C + ------------------ - 3*\/ 3 *atan|-----------------| | 2 2 \ 3 / | x + 7*x + 13 | /
$$\int \frac{x - 1}{\left(x^{2} + 7 x\right) + 13}\, dx = C + \frac{\log{\left(x^{2} + 7 x + 13 \right)}}{2} - 3 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x + \frac{7}{2}\right)}{3} \right)}$$
Gráfica
Respuesta
[src]
/ ___\ log(21) log(13) ___ / ___\ ___ |7*\/ 3 | ------- - ------- - 3*\/ 3 *atan\3*\/ 3 / + 3*\/ 3 *atan|-------| 2 2 \ 3 /
$$- 3 \sqrt{3} \operatorname{atan}{\left(3 \sqrt{3} \right)} - \frac{\log{\left(13 \right)}}{2} + \frac{\log{\left(21 \right)}}{2} + 3 \sqrt{3} \operatorname{atan}{\left(\frac{7 \sqrt{3}}{3} \right)}$$
=
=
/ ___\ log(21) log(13) ___ / ___\ ___ |7*\/ 3 | ------- - ------- - 3*\/ 3 *atan\3*\/ 3 / + 3*\/ 3 *atan|-------| 2 2 \ 3 /
$$- 3 \sqrt{3} \operatorname{atan}{\left(3 \sqrt{3} \right)} - \frac{\log{\left(13 \right)}}{2} + \frac{\log{\left(21 \right)}}{2} + 3 \sqrt{3} \operatorname{atan}{\left(\frac{7 \sqrt{3}}{3} \right)}$$
log(21)/2 - log(13)/2 - 3*sqrt(3)*atan(3*sqrt(3)) + 3*sqrt(3)*atan(7*sqrt(3)/3)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.