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