Sr Examen
Integral de (x+4)/(2x^2-6x+8) dx
Solución
Ha introducido
[src]
1 / | | x + 4 | -------------- dx | 2 | 2*x - 6*x + 8 | / 0
$$\int\limits_{0}^{1} \frac{x + 4}{\left(2 x^{2} - 6 x\right) + 8}\, dx$$
Integral((x + 4)/(2*x^2 - 6*x + 8), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 4 | -------------- dx | 2 | 2*x - 6*x + 8 | /
Reescribimos la función subintegral
/ 2*2*x - 6 \
|--------------| / 11 \
| 2 | |-----|
x + 4 \2*x - 6*x + 8/ \2*7/2/
-------------- = ---------------- + ---------------------------
2 4 2
2*x - 6*x + 8 / ___ ___\
|-2*\/ 7 3*\/ 7 |
|--------*x + -------| + 1
\ 7 7 / o
/ | | x + 4 | -------------- dx | 2 = | 2*x - 6*x + 8 | /
/
|
| 1
/ 11* | --------------------------- dx
| | 2
| 2*2*x - 6 | / ___ ___\
| -------------- dx | |-2*\/ 7 3*\/ 7 |
| 2 | |--------*x + -------| + 1
| 2*x - 6*x + 8 | \ 7 7 /
| |
/ /
-------------------- + ------------------------------------
4 7 En integral
/
|
| 2*2*x - 6
| -------------- dx
| 2
| 2*x - 6*x + 8
|
/
--------------------
4 hacemos el cambio
2 u = -6*x + 2*x
entonces
integral =
/
|
| 1
| ----- du
| 8 + u
|
/ log(8 + u)
----------- = ----------
4 4 hacemos cambio inverso
/
|
| 2*2*x - 6
| -------------- dx
| 2
| 2*x - 6*x + 8
| / 2 \
/ log\4 + x - 3*x/
-------------------- = -----------------
4 4 En integral
/
|
| 1
11* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 3*\/ 7 |
| |--------*x + -------| + 1
| \ 7 7 /
|
/
------------------------------------
7 hacemos el cambio
___ ___
3*\/ 7 2*x*\/ 7
v = ------- - ---------
7 7 entonces
integral =
/
|
| 1
11* | ------ dv
| 2
| 1 + v
|
/ 11*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
11* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 3*\/ 7 |
| |--------*x + -------| + 1 / ___ ___\
| \ 7 7 / ___ | 3*\/ 7 2*x*\/ 7 |
| 11*\/ 7 *atan|- ------- + ---------|
/ \ 7 7 /
------------------------------------ = ------------------------------------
7 14 La solución:
/ ___ ___\
___ | 3*\/ 7 2*x*\/ 7 |
/ 2 \ 11*\/ 7 *atan|- ------- + ---------|
log\4 + x - 3*x/ \ 7 7 /
C + ----------------- + ------------------------------------
4 14
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(-3/2 + x)| | / 2 \ 11*\/ 7 *atan|------------------| | x + 4 log\4 + x - 3*x/ \ 7 / | -------------- dx = C + ----------------- + --------------------------------- | 2 4 14 | 2*x - 6*x + 8 | /
$$\int \frac{x + 4}{\left(2 x^{2} - 6 x\right) + 8}\, dx = C + \frac{\log{\left(x^{2} - 3 x + 4 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x - \frac{3}{2}\right)}{7} \right)}}{14}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
11*\/ 7 *atan|-----| 11*\/ 7 *atan|-------|
log(4) log(2) \ 7 / \ 7 /
- ------ + ------ - -------------------- + ----------------------
4 4 14 14
$$- \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{14} - \frac{\log{\left(4 \right)}}{4} + \frac{\log{\left(2 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{14}$$
=
=
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
11*\/ 7 *atan|-----| 11*\/ 7 *atan|-------|
log(4) log(2) \ 7 / \ 7 /
- ------ + ------ - -------------------- + ----------------------
4 4 14 14
$$- \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{14} - \frac{\log{\left(4 \right)}}{4} + \frac{\log{\left(2 \right)}}{4} + \frac{11 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{14}$$
-log(4)/4 + log(2)/4 - 11*sqrt(7)*atan(sqrt(7)/7)/14 + 11*sqrt(7)*atan(3*sqrt(7)/7)/14
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.