Sr Examen
Integral de (19-x)*dx/(x^2+6*x+23) dx
Solución
Ha introducido
[src]
1 / | | 19 - x | ------------- dx | 2 | x + 6*x + 23 | / 0
$$\int\limits_{0}^{1} \frac{19 - x}{\left(x^{2} + 6 x\right) + 23}\, dx$$
Integral((19 - x)/(x^2 + 6*x + 23), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 19 - x | ------------- dx | 2 | x + 6*x + 23 | /
Reescribimos la función subintegral
/ 2*x + 6 \
|-------------| /22\
| 2 | |--|
19 - x \x + 6*x + 23/ \14/
------------- = - --------------- + ----------------------------
2 2 2
x + 6*x + 23 / ____ ____\
|-\/ 14 3*\/ 14 |
|--------*x - --------| + 1
\ 14 14 / o
/ | | 19 - x | ------------- dx | 2 = | x + 6*x + 23 | /
/
|
| 1
/ 11* | ---------------------------- dx
| | 2
| 2*x + 6 | / ____ ____\
| ------------- dx | |-\/ 14 3*\/ 14 |
| 2 | |--------*x - --------| + 1
| x + 6*x + 23 | \ 14 14 /
| |
/ /
- ------------------- + -------------------------------------
2 7 En integral
/
|
| 2*x + 6
- | ------------- dx
| 2
| x + 6*x + 23
|
/
---------------------
2 hacemos el cambio
2 u = x + 6*x
entonces
integral =
/
|
| 1
- | ------ du
| 23 + u
|
/ -log(23 + u)
-------------- = -------------
2 2 hacemos cambio inverso
/
|
| 2*x + 6
- | ------------- dx
| 2
| x + 6*x + 23
| / 2 \
/ -log\23 + x + 6*x/
--------------------- = --------------------
2 2 En integral
/
|
| 1
11* | ---------------------------- dx
| 2
| / ____ ____\
| |-\/ 14 3*\/ 14 |
| |--------*x - --------| + 1
| \ 14 14 /
|
/
-------------------------------------
7 hacemos el cambio
____ ____
3*\/ 14 x*\/ 14
v = - -------- - --------
14 14 entonces
integral =
/
|
| 1
11* | ------ dv
| 2
| 1 + v
|
/ 11*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
11* | ---------------------------- dx
| 2
| / ____ ____\
| |-\/ 14 3*\/ 14 |
| |--------*x - --------| + 1 / ____ ____\
| \ 14 14 / ____ |3*\/ 14 x*\/ 14 |
| 11*\/ 14 *atan|-------- + --------|
/ \ 14 14 /
------------------------------------- = -----------------------------------
7 7 La solución:
/ ____ ____\
____ |3*\/ 14 x*\/ 14 |
/ 2 \ 11*\/ 14 *atan|-------- + --------|
log\23 + x + 6*x/ \ 14 14 /
C - ------------------ + -----------------------------------
2 7
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |\/ 14 *(3 + x)| | / 2 \ 11*\/ 14 *atan|--------------| | 19 - x log\23 + x + 6*x/ \ 14 / | ------------- dx = C - ------------------ + ------------------------------ | 2 2 7 | x + 6*x + 23 | /
$$\int \frac{19 - x}{\left(x^{2} + 6 x\right) + 23}\, dx = C - \frac{\log{\left(x^{2} + 6 x + 23 \right)}}{2} + \frac{11 \sqrt{14} \operatorname{atan}{\left(\frac{\sqrt{14} \left(x + 3\right)}{14} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 14 | ____ |2*\/ 14 |
11*\/ 14 *atan|--------| 11*\/ 14 *atan|--------|
log(23) log(30) \ 14 / \ 7 /
------- - ------- - ------------------------ + ------------------------
2 2 7 7
$$- \frac{11 \sqrt{14} \operatorname{atan}{\left(\frac{3 \sqrt{14}}{14} \right)}}{7} - \frac{\log{\left(30 \right)}}{2} + \frac{\log{\left(23 \right)}}{2} + \frac{11 \sqrt{14} \operatorname{atan}{\left(\frac{2 \sqrt{14}}{7} \right)}}{7}$$
=
=
/ ____\ / ____\
____ |3*\/ 14 | ____ |2*\/ 14 |
11*\/ 14 *atan|--------| 11*\/ 14 *atan|--------|
log(23) log(30) \ 14 / \ 7 /
------- - ------- - ------------------------ + ------------------------
2 2 7 7
$$- \frac{11 \sqrt{14} \operatorname{atan}{\left(\frac{3 \sqrt{14}}{14} \right)}}{7} - \frac{\log{\left(30 \right)}}{2} + \frac{\log{\left(23 \right)}}{2} + \frac{11 \sqrt{14} \operatorname{atan}{\left(\frac{2 \sqrt{14}}{7} \right)}}{7}$$
log(23)/2 - log(30)/2 - 11*sqrt(14)*atan(3*sqrt(14)/14)/7 + 11*sqrt(14)*atan(2*sqrt(14)/7)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.