Sr Examen
Integral de (x-7)/(x^2+x+6) dx
Solución
Ha introducido
[src]
1 / | | x - 7 | ---------- dx | 2 | x + x + 6 | / 0
$$\int\limits_{0}^{1} \frac{x - 7}{\left(x^{2} + x\right) + 6}\, dx$$
Integral((x - 7)/(x^2 + x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x - 7 | ---------- dx | 2 | x + x + 6 | /
Reescribimos la función subintegral
/ 2*x + 1 \
|----------| / -15 \
| 2 | |------|
x - 7 \x + x + 6/ \2*23/4/
---------- = ------------ + ---------------------------
2 2 2
x + x + 6 / ____ ____\
|-2*\/ 23 \/ 23 |
|---------*x - ------| + 1
\ 23 23 / o
/ | | x - 7 | ---------- dx | 2 = | x + x + 6 | /
/
|
| 1
/ 30* | --------------------------- dx
| | 2
| 2*x + 1 | / ____ ____\
| ---------- dx | |-2*\/ 23 \/ 23 |
| 2 | |---------*x - ------| + 1
| x + x + 6 | \ 23 23 /
| |
/ /
---------------- - ------------------------------------
2 23 En integral
/
|
| 2*x + 1
| ---------- dx
| 2
| x + x + 6
|
/
----------------
2 hacemos el cambio
2 u = x + x
entonces
integral =
/
|
| 1
| ----- du
| 6 + u
|
/ log(6 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x + 1
| ---------- dx
| 2
| x + x + 6
| / 2\
/ log\6 + x + x /
---------------- = ---------------
2 2 En integral
/
|
| 1
-30* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 23 \/ 23 |
| |---------*x - ------| + 1
| \ 23 23 /
|
/
-------------------------------------
23 hacemos el cambio
____ ____
\/ 23 2*x*\/ 23
v = - ------ - ----------
23 23 entonces
integral =
/
|
| 1
-30* | ------ dv
| 2
| 1 + v
|
/ -30*atan(v)
---------------- = -----------
23 23 hacemos cambio inverso
/
|
| 1
-30* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 23 \/ 23 |
| |---------*x - ------| + 1 / ____ ____\
| \ 23 23 / ____ |\/ 23 2*x*\/ 23 |
| -15*\/ 23 *atan|------ + ----------|
/ \ 23 23 /
------------------------------------- = ------------------------------------
23 23 La solución:
/ ____ ____\
____ |\/ 23 2*x*\/ 23 |
/ 2\ 15*\/ 23 *atan|------ + ----------|
log\6 + x + x / \ 23 23 /
C + --------------- - -----------------------------------
2 23
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 23 *(1/2 + x)| | / 2\ 15*\/ 23 *atan|------------------| | x - 7 log\6 + x + x / \ 23 / | ---------- dx = C + --------------- - ---------------------------------- | 2 2 23 | x + x + 6 | /
$$\int \frac{x - 7}{\left(x^{2} + x\right) + 6}\, dx = C + \frac{\log{\left(x^{2} + x + 6 \right)}}{2} - \frac{15 \sqrt{23} \operatorname{atan}{\left(\frac{2 \sqrt{23} \left(x + \frac{1}{2}\right)}{23} \right)}}{23}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 23 | ____ |\/ 23 |
15*\/ 23 *atan|--------| 15*\/ 23 *atan|------|
log(8) log(6) \ 23 / \ 23 /
------ - ------ - ------------------------ + ----------------------
2 2 23 23
$$- \frac{15 \sqrt{23} \operatorname{atan}{\left(\frac{3 \sqrt{23}}{23} \right)}}{23} - \frac{\log{\left(6 \right)}}{2} + \frac{15 \sqrt{23} \operatorname{atan}{\left(\frac{\sqrt{23}}{23} \right)}}{23} + \frac{\log{\left(8 \right)}}{2}$$
=
=
/ ____\ / ____\
____ |3*\/ 23 | ____ |\/ 23 |
15*\/ 23 *atan|--------| 15*\/ 23 *atan|------|
log(8) log(6) \ 23 / \ 23 /
------ - ------ - ------------------------ + ----------------------
2 2 23 23
$$- \frac{15 \sqrt{23} \operatorname{atan}{\left(\frac{3 \sqrt{23}}{23} \right)}}{23} - \frac{\log{\left(6 \right)}}{2} + \frac{15 \sqrt{23} \operatorname{atan}{\left(\frac{\sqrt{23}}{23} \right)}}{23} + \frac{\log{\left(8 \right)}}{2}$$
log(8)/2 - log(6)/2 - 15*sqrt(23)*atan(3*sqrt(23)/23)/23 + 15*sqrt(23)*atan(sqrt(23)/23)/23
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.