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