Sr Examen
Integral de (x)/(x^2+7x+13) dx
Solución
Ha introducido
[src]
1 / | | x | ------------- dx | 2 | x + 7*x + 13 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 7 x\right) + 13}\, dx$$
Integral(x/(x^2 + 7*x + 13), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------- dx | 2 | x + 7*x + 13 | /
Reescribimos la función subintegral
/ 2*x + 7 \
|-------------| / -7 \
| 2 | |-----|
x \x + 7*x + 13/ \2*3/4/
------------- = --------------- + ---------------------------
2 2 2
x + 7*x + 13 / ___ ___\
|-2*\/ 3 7*\/ 3 |
|--------*x - -------| + 1
\ 3 3 / o
/ | | x | ------------- dx | 2 = | x + 7*x + 13 | /
/
|
| 1
/ 14* | --------------------------- dx
| | 2
| 2*x + 7 | / ___ ___\
| ------------- dx | |-2*\/ 3 7*\/ 3 |
| 2 | |--------*x - -------| + 1
| x + 7*x + 13 | \ 3 3 /
| |
/ /
------------------- - ------------------------------------
2 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
-14* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1
| \ 3 3 /
|
/
-------------------------------------
3 hacemos el cambio
___ ___
7*\/ 3 2*x*\/ 3
v = - ------- - ---------
3 3 entonces
integral =
/
|
| 1
-14* | ------ dv
| 2
| 1 + v
|
/ -14*atan(v)
---------------- = -----------
3 3 hacemos cambio inverso
/
|
| 1
-14* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 3 7*\/ 3 |
| |--------*x - -------| + 1 / ___ ___\
| \ 3 3 / ___ |7*\/ 3 2*x*\/ 3 |
| -7*\/ 3 *atan|------- + ---------|
/ \ 3 3 /
------------------------------------- = ----------------------------------
3 3 La solución:
/ ___ ___\
___ |7*\/ 3 2*x*\/ 3 |
/ 2 \ 7*\/ 3 *atan|------- + ---------|
log\13 + x + 7*x/ \ 3 3 /
C + ------------------ - ---------------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 3 *(7/2 + x)| | / 2 \ 7*\/ 3 *atan|-----------------| | x log\13 + x + 7*x/ \ 3 / | ------------- dx = C + ------------------ - ------------------------------- | 2 2 3 | x + 7*x + 13 | /
$$\int \frac{x}{\left(x^{2} + 7 x\right) + 13}\, dx = C + \frac{\log{\left(x^{2} + 7 x + 13 \right)}}{2} - \frac{7 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x + \frac{7}{2}\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |7*\/ 3 |
___ / ___\ 7*\/ 3 *atan|-------|
log(21) log(13) 7*\/ 3 *atan\3*\/ 3 / \ 3 /
------- - ------- - --------------------- + ---------------------
2 2 3 3
$$- \frac{7 \sqrt{3} \operatorname{atan}{\left(3 \sqrt{3} \right)}}{3} - \frac{\log{\left(13 \right)}}{2} + \frac{\log{\left(21 \right)}}{2} + \frac{7 \sqrt{3} \operatorname{atan}{\left(\frac{7 \sqrt{3}}{3} \right)}}{3}$$
=
=
/ ___\
___ |7*\/ 3 |
___ / ___\ 7*\/ 3 *atan|-------|
log(21) log(13) 7*\/ 3 *atan\3*\/ 3 / \ 3 /
------- - ------- - --------------------- + ---------------------
2 2 3 3
$$- \frac{7 \sqrt{3} \operatorname{atan}{\left(3 \sqrt{3} \right)}}{3} - \frac{\log{\left(13 \right)}}{2} + \frac{\log{\left(21 \right)}}{2} + \frac{7 \sqrt{3} \operatorname{atan}{\left(\frac{7 \sqrt{3}}{3} \right)}}{3}$$
log(21)/2 - log(13)/2 - 7*sqrt(3)*atan(3*sqrt(3))/3 + 7*sqrt(3)*atan(7*sqrt(3)/3)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.