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