Sr Examen
Integral de (2*x+3)/(x^2-7*x+14) dx
Solución
Ha introducido
[src]
1 / | | 2*x + 3 | ------------- dx | 2 | x - 7*x + 14 | / 0
$$\int\limits_{0}^{1} \frac{2 x + 3}{\left(x^{2} - 7 x\right) + 14}\, dx$$
Integral((2*x + 3)/(x^2 - 7*x + 14), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x + 3 | ------------- dx | 2 | x - 7*x + 14 | /
Reescribimos la función subintegral
/ 10\
|---|
2*x + 3 2*x - 7 \7/4/
------------- = ------------- + -------------------------
2 2 2
x - 7*x + 14 x - 7*x + 14 / ___ \
|-2*\/ 7 ___|
|--------*x + \/ 7 | + 1
\ 7 / o
/ | | 2*x + 3 | ------------- dx | 2 = | x - 7*x + 14 | /
/
|
| 1
40* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 7 ___|
| |--------*x + \/ 7 | + 1
| \ 7 / /
| |
/ | 2*x - 7
---------------------------------- + | ------------- dx
7 | 2
| x - 7*x + 14
|
/ En integral
/ | | 2*x - 7 | ------------- dx | 2 | x - 7*x + 14 | /
hacemos el cambio
2 u = x - 7*x
entonces
integral =
/ | | 1 | ------ du = log(14 + u) | 14 + u | /
hacemos cambio inverso
/ | | 2*x - 7 / 2 \ | ------------- dx = log\14 + x - 7*x/ | 2 | x - 7*x + 14 | /
En integral
/
|
| 1
40* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 7 ___|
| |--------*x + \/ 7 | + 1
| \ 7 /
|
/
----------------------------------
7 hacemos el cambio
___
___ 2*x*\/ 7
v = \/ 7 - ---------
7 entonces
integral =
/
|
| 1
40* | ------ dv
| 2
| 1 + v
|
/ 40*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
40* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 7 ___|
| |--------*x + \/ 7 | + 1 / ___\
| \ 7 / ___ | ___ 2*x*\/ 7 |
| 20*\/ 7 *atan|- \/ 7 + ---------|
/ \ 7 /
---------------------------------- = ----------------------------------
7 7 La solución:
/ ___\
___ | ___ 2*x*\/ 7 |
20*\/ 7 *atan|- \/ 7 + ---------|
\ 7 / / 2 \
C + ---------------------------------- + log\14 + x - 7*x/
7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(-7/2 + x)| | 20*\/ 7 *atan|------------------| | 2*x + 3 \ 7 / / 2 \ | ------------- dx = C + --------------------------------- + log\14 + x - 7*x/ | 2 7 | x - 7*x + 14 | /
$$\int \frac{2 x + 3}{\left(x^{2} - 7 x\right) + 14}\, dx = C + \log{\left(x^{2} - 7 x + 14 \right)} + \frac{20 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x - \frac{7}{2}\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\
___ |5*\/ 7 |
20*\/ 7 *atan|-------| ___ / ___\
\ 7 / 20*\/ 7 *atan\\/ 7 /
-log(14) - ---------------------- + -------------------- + log(8)
7 7
$$- \frac{20 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7} - \log{\left(14 \right)} + \log{\left(8 \right)} + \frac{20 \sqrt{7} \operatorname{atan}{\left(\sqrt{7} \right)}}{7}$$
=
=
/ ___\
___ |5*\/ 7 |
20*\/ 7 *atan|-------| ___ / ___\
\ 7 / 20*\/ 7 *atan\\/ 7 /
-log(14) - ---------------------- + -------------------- + log(8)
7 7
$$- \frac{20 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7} - \log{\left(14 \right)} + \log{\left(8 \right)} + \frac{20 \sqrt{7} \operatorname{atan}{\left(\sqrt{7} \right)}}{7}$$
-log(14) - 20*sqrt(7)*atan(5*sqrt(7)/7)/7 + 20*sqrt(7)*atan(sqrt(7))/7 + log(8)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.