Sr Examen
Integral de (2*x+5)/(2*x^2+3*x+5) dx
Solución
Ha introducido
[src]
1 / | | 2*x + 5 | -------------- dx | 2 | 2*x + 3*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{2 x + 5}{\left(2 x^{2} + 3 x\right) + 5}\, dx$$
Integral((2*x + 5)/(2*x^2 + 3*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x + 5 | -------------- dx | 2 | 2*x + 3*x + 5 | /
Reescribimos la función subintegral
/ 2*2*x + 3 \
|--------------| / 7 \
| 2 | |------|
2*x + 5 \2*x + 3*x + 5/ \2*31/8/
-------------- = ---------------- + -----------------------------
2 2 2
2*x + 3*x + 5 / ____ ____\
|-4*\/ 31 3*\/ 31 |
|---------*x - --------| + 1
\ 31 31 / o
/ | | 2*x + 5 | -------------- dx | 2 = | 2*x + 3*x + 5 | /
/
|
| 1
/ 28* | ----------------------------- dx
| | 2
| 2*2*x + 3 | / ____ ____\
| -------------- dx | |-4*\/ 31 3*\/ 31 |
| 2 | |---------*x - --------| + 1
| 2*x + 3*x + 5 | \ 31 31 /
| |
/ /
-------------------- + --------------------------------------
2 31 En integral
/
|
| 2*2*x + 3
| -------------- dx
| 2
| 2*x + 3*x + 5
|
/
--------------------
2 hacemos el cambio
2 u = 2*x + 3*x
entonces
integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*2*x + 3
| -------------- dx
| 2
| 2*x + 3*x + 5
| / 2 \
/ log\5 + 2*x + 3*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
28* | ----------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 31 3*\/ 31 |
| |---------*x - --------| + 1
| \ 31 31 /
|
/
--------------------------------------
31 hacemos el cambio
____ ____
3*\/ 31 4*x*\/ 31
v = - -------- - ----------
31 31 entonces
integral =
/
|
| 1
28* | ------ dv
| 2
| 1 + v
|
/ 28*atan(v)
--------------- = ----------
31 31 hacemos cambio inverso
/
|
| 1
28* | ----------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 31 3*\/ 31 |
| |---------*x - --------| + 1 / ____ ____\
| \ 31 31 / ____ |3*\/ 31 4*x*\/ 31 |
| 7*\/ 31 *atan|-------- + ----------|
/ \ 31 31 /
-------------------------------------- = ------------------------------------
31 31 La solución:
/ ____ ____\
/5 2 3*x\ ____ |3*\/ 31 4*x*\/ 31 |
log|- + x + ---| 7*\/ 31 *atan|-------- + ----------|
\2 2 / \ 31 31 /
C + ----------------- + ------------------------------------
2 31
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |4*\/ 31 *(3/4 + x)| | / 2 \ 7*\/ 31 *atan|------------------| | 2*x + 5 log\5 + 2*x + 3*x/ \ 31 / | -------------- dx = C + ------------------- + --------------------------------- | 2 2 31 | 2*x + 3*x + 5 | /
$$\int \frac{2 x + 5}{\left(2 x^{2} + 3 x\right) + 5}\, dx = C + \frac{\log{\left(2 x^{2} + 3 x + 5 \right)}}{2} + \frac{7 \sqrt{31} \operatorname{atan}{\left(\frac{4 \sqrt{31} \left(x + \frac{3}{4}\right)}{31} \right)}}{31}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 31 | ____ |7*\/ 31 |
7*\/ 31 *atan|--------| 7*\/ 31 *atan|--------|
log(5) log(5/2) \ 31 / \ 31 /
------ - -------- - ----------------------- + -----------------------
2 2 31 31
$$- \frac{7 \sqrt{31} \operatorname{atan}{\left(\frac{3 \sqrt{31}}{31} \right)}}{31} - \frac{\log{\left(\frac{5}{2} \right)}}{2} + \frac{\log{\left(5 \right)}}{2} + \frac{7 \sqrt{31} \operatorname{atan}{\left(\frac{7 \sqrt{31}}{31} \right)}}{31}$$
=
=
/ ____\ / ____\
____ |3*\/ 31 | ____ |7*\/ 31 |
7*\/ 31 *atan|--------| 7*\/ 31 *atan|--------|
log(5) log(5/2) \ 31 / \ 31 /
------ - -------- - ----------------------- + -----------------------
2 2 31 31
$$- \frac{7 \sqrt{31} \operatorname{atan}{\left(\frac{3 \sqrt{31}}{31} \right)}}{31} - \frac{\log{\left(\frac{5}{2} \right)}}{2} + \frac{\log{\left(5 \right)}}{2} + \frac{7 \sqrt{31} \operatorname{atan}{\left(\frac{7 \sqrt{31}}{31} \right)}}{31}$$
log(5)/2 - log(5/2)/2 - 7*sqrt(31)*atan(3*sqrt(31)/31)/31 + 7*sqrt(31)*atan(7*sqrt(31)/31)/31
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.