Sr Examen
Integral de (4x-5)/(4x^2+4x+10) dx
Solución
Ha introducido
[src]
1 / | | 4*x - 5 | --------------- dx | 2 | 4*x + 4*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{4 x - 5}{\left(4 x^{2} + 4 x\right) + 10}\, dx$$
Integral((4*x - 5)/(4*x^2 + 4*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 4*x - 5 | --------------- dx | 2 | 4*x + 4*x + 10 | /
Reescribimos la función subintegral
/ 4*2*x + 4 \
|---------------| /-7 \
| 2 | |---|
4*x - 5 \4*x + 4*x + 10/ \ 9 /
--------------- = ----------------- + ----------------
2 2 2
4*x + 4*x + 10 / 2*x 1\
|- --- - -| + 1
\ 3 3/ o
/ | | 4*x - 5 | --------------- dx | 2 = | 4*x + 4*x + 10 | /
/
|
/ | 1
| 7* | ---------------- dx
| 4*2*x + 4 | 2
| --------------- dx | / 2*x 1\
| 2 | |- --- - -| + 1
| 4*x + 4*x + 10 | \ 3 3/
| |
/ /
--------------------- - ------------------------
2 9 En integral
/
|
| 4*2*x + 4
| --------------- dx
| 2
| 4*x + 4*x + 10
|
/
---------------------
2 hacemos el cambio
2 u = 4*x + 4*x
entonces
integral =
/
|
| 1
| ------ du
| 10 + u
|
/ log(10 + u)
------------ = -----------
2 2 hacemos cambio inverso
/
|
| 4*2*x + 4
| --------------- dx
| 2
| 4*x + 4*x + 10
| / 2\
/ log\5 + 2*x + 2*x /
--------------------- = -------------------
2 2 En integral
/
|
| 1
-7* | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/
|
/
-------------------------
9 hacemos el cambio
1 2*x
v = - - - ---
3 3 entonces
integral =
/
|
| 1
-7* | ------ dv
| 2
| 1 + v
|
/ -7*atan(v)
--------------- = ----------
9 9 hacemos cambio inverso
/
|
| 1
-7* | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/ /1 2*x\
| -7*atan|- + ---|
/ \3 3 /
------------------------- = ----------------
9 6 La solución:
/5 2\ /1 2*x\
log|- + x + x | 7*atan|- + ---|
\2 / \3 3 /
C + --------------- - ---------------
2 6
Respuesta (Indefinida)
[src]
/ /1 2*x\ | / 2\ 7*atan|- + ---| | 4*x - 5 log\5 + 2*x + 2*x / \3 3 / | --------------- dx = C + ------------------- - --------------- | 2 2 6 | 4*x + 4*x + 10 | /
$$\int \frac{4 x - 5}{\left(4 x^{2} + 4 x\right) + 10}\, dx = C + \frac{\log{\left(2 x^{2} + 2 x + 5 \right)}}{2} - \frac{7 \operatorname{atan}{\left(\frac{2 x}{3} + \frac{1}{3} \right)}}{6}$$
Gráfica
Respuesta
[src]
log(9/2) 7*pi log(5/2) 7*atan(1/3) -------- - ---- - -------- + ----------- 2 24 2 6
$$- \frac{7 \pi}{24} - \frac{\log{\left(\frac{5}{2} \right)}}{2} + \frac{7 \operatorname{atan}{\left(\frac{1}{3} \right)}}{6} + \frac{\log{\left(\frac{9}{2} \right)}}{2}$$
=
=
log(9/2) 7*pi log(5/2) 7*atan(1/3) -------- - ---- - -------- + ----------- 2 24 2 6
$$- \frac{7 \pi}{24} - \frac{\log{\left(\frac{5}{2} \right)}}{2} + \frac{7 \operatorname{atan}{\left(\frac{1}{3} \right)}}{6} + \frac{\log{\left(\frac{9}{2} \right)}}{2}$$
log(9/2)/2 - 7*pi/24 - log(5/2)/2 + 7*atan(1/3)/6
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.