Sr Examen
Integral de (x+5)/(2*x^2+4*x+3) dx
Solución
Ha introducido
[src]
1 / | | x + 5 | -------------- dx | 2 | 2*x + 4*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{x + 5}{\left(2 x^{2} + 4 x\right) + 3}\, dx$$
Integral((x + 5)/(2*x^2 + 4*x + 3), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 5 | -------------- dx | 2 | 2*x + 4*x + 3 | /
Reescribimos la función subintegral
/ 2*2*x + 4 \
|--------------| /4\
| 2 | |-|
x + 5 \2*x + 4*x + 3/ \1/
-------------- = ---------------- + -----------------------
2 4 2
2*x + 4*x + 3 / ___ ___\
\-\/ 2 *x - \/ 2 / + 1o
/ | | x + 5 | -------------- dx | 2 = | 2*x + 4*x + 3 | /
/
|
| 2*2*x + 4
| -------------- dx
| 2
/ | 2*x + 4*x + 3
| |
| 1 /
4* | ----------------------- 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 4* | ----------------------- dx | 2 | / ___ ___\ | \-\/ 2 *x - \/ 2 / + 1 | /
hacemos el cambio
___ ___ v = - \/ 2 - x*\/ 2
entonces
integral =
/ | | 1 4* | ------ dv = 4*atan(v) | 2 | 1 + v | /
hacemos cambio inverso
/ | | 1 ___ / ___ ___\ 4* | ----------------------- dx = 2*\/ 2 *atan\\/ 2 + x*\/ 2 / | 2 | / ___ ___\ | \-\/ 2 *x - \/ 2 / + 1 | /
La solución:
/3 2 \
log|- + x + 2*x|
\2 / ___ / ___ ___\
C + ----------------- + 2*\/ 2 *atan\\/ 2 + x*\/ 2 /
4
Respuesta (Indefinida)
[src]
/ | / 2 \ | x + 5 log\3 + 2*x + 4*x/ ___ / ___ \ | -------------- dx = C + ------------------- + 2*\/ 2 *atan\\/ 2 *(1 + x)/ | 2 4 | 2*x + 4*x + 3 | /
$$\int \frac{x + 5}{\left(2 x^{2} + 4 x\right) + 3}\, dx = C + \frac{\log{\left(2 x^{2} + 4 x + 3 \right)}}{4} + 2 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \left(x + 1\right) \right)}$$
Gráfica
Respuesta
[src]
log(3/2) log(9/2) ___ / ___\ ___ / ___\
- -------- + -------- - 2*\/ 2 *atan\\/ 2 / + 2*\/ 2 *atan\2*\/ 2 /
4 4
$$- 2 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} - \frac{\log{\left(\frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{9}{2} \right)}}{4} + 2 \sqrt{2} \operatorname{atan}{\left(2 \sqrt{2} \right)}$$
=
=
log(3/2) log(9/2) ___ / ___\ ___ / ___\
- -------- + -------- - 2*\/ 2 *atan\\/ 2 / + 2*\/ 2 *atan\2*\/ 2 /
4 4
$$- 2 \sqrt{2} \operatorname{atan}{\left(\sqrt{2} \right)} - \frac{\log{\left(\frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{9}{2} \right)}}{4} + 2 \sqrt{2} \operatorname{atan}{\left(2 \sqrt{2} \right)}$$
-log(3/2)/4 + log(9/2)/4 - 2*sqrt(2)*atan(sqrt(2)) + 2*sqrt(2)*atan(2*sqrt(2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.