Sr Examen
Integral de (5x-1)/(x^(2)+3x+3) dx
Solución
Ha introducido
[src]
1 / | | 5*x - 1 | ------------ dx | 2 | x + 3*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{5 x - 1}{\left(x^{2} + 3 x\right) + 3}\, dx$$
Integral((5*x - 1)/(x^2 + 3*x + 3), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 5*x - 1 | ------------ dx | 2 | x + 3*x + 3 | /
Reescribimos la función subintegral
2*x + 3
5*------------ / -17 \
2 |-----|
5*x - 1 x + 3*x + 3 \2*3/4/
------------ = -------------- + -------------------------
2 2 2
x + 3*x + 3 / ___ \
|-2*\/ 3 ___|
|--------*x - \/ 3 | + 1
\ 3 / o
/ | | 5*x - 1 | ------------ dx | 2 = | x + 3*x + 3 | /
/
|
| 1
34* | ------------------------- dx /
| 2 |
| / ___ \ | 2*x + 3
| |-2*\/ 3 ___| 5* | ------------ dx
| |--------*x - \/ 3 | + 1 | 2
| \ 3 / | x + 3*x + 3
| |
/ /
- ---------------------------------- + --------------------
3 2 En integral
/
|
| 2*x + 3
5* | ------------ dx
| 2
| x + 3*x + 3
|
/
--------------------
2 hacemos el cambio
2 u = x + 3*x
entonces
integral =
/
|
| 1
5* | ----- du
| 3 + u
|
/ 5*log(3 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 3
5* | ------------ dx
| 2
| x + 3*x + 3
| / 2 \
/ 5*log\3 + x + 3*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
-34* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 3 ___|
| |--------*x - \/ 3 | + 1
| \ 3 /
|
/
-----------------------------------
3 hacemos el cambio
___
___ 2*x*\/ 3
v = - \/ 3 - ---------
3 entonces
integral =
/
|
| 1
-34* | ------ dv
| 2
| 1 + v
|
/ -34*atan(v)
---------------- = -----------
3 3 hacemos cambio inverso
/
|
| 1
-34* | ------------------------- dx
| 2
| / ___ \
| |-2*\/ 3 ___|
| |--------*x - \/ 3 | + 1 / ___\
| \ 3 / ___ | ___ 2*x*\/ 3 |
| -17*\/ 3 *atan|\/ 3 + ---------|
/ \ 3 /
----------------------------------- = ---------------------------------
3 3 La solución:
/ ___\
___ | ___ 2*x*\/ 3 |
/ 2 \ 17*\/ 3 *atan|\/ 3 + ---------|
5*log\3 + x + 3*x/ \ 3 /
C + ------------------- - --------------------------------
2 3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 3 *(3/2 + x)| | / 2 \ 17*\/ 3 *atan|-----------------| | 5*x - 1 5*log\3 + x + 3*x/ \ 3 / | ------------ dx = C + ------------------- - -------------------------------- | 2 2 3 | x + 3*x + 3 | /
$$\int \frac{5 x - 1}{\left(x^{2} + 3 x\right) + 3}\, dx = C + \frac{5 \log{\left(x^{2} + 3 x + 3 \right)}}{2} - \frac{17 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} \left(x + \frac{3}{2}\right)}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\
___ |5*\/ 3 |
17*\/ 3 *atan|-------| ___
5*log(3) 5*log(7) \ 3 / 17*pi*\/ 3
- -------- + -------- - ---------------------- + -----------
2 2 3 9
$$- \frac{17 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3} - \frac{5 \log{\left(3 \right)}}{2} + \frac{5 \log{\left(7 \right)}}{2} + \frac{17 \sqrt{3} \pi}{9}$$
=
=
/ ___\
___ |5*\/ 3 |
17*\/ 3 *atan|-------| ___
5*log(3) 5*log(7) \ 3 / 17*pi*\/ 3
- -------- + -------- - ---------------------- + -----------
2 2 3 9
$$- \frac{17 \sqrt{3} \operatorname{atan}{\left(\frac{5 \sqrt{3}}{3} \right)}}{3} - \frac{5 \log{\left(3 \right)}}{2} + \frac{5 \log{\left(7 \right)}}{2} + \frac{17 \sqrt{3} \pi}{9}$$
-5*log(3)/2 + 5*log(7)/2 - 17*sqrt(3)*atan(5*sqrt(3)/3)/3 + 17*pi*sqrt(3)/9
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.