Sr Examen
Integral de (-4x+19)/((3x^2)+6x+19) dx
Solución
Ha introducido
[src]
1 / | | -4*x + 19 | --------------- dx | 2 | 3*x + 6*x + 19 | / 0
$$\int\limits_{0}^{1} \frac{19 - 4 x}{\left(3 x^{2} + 6 x\right) + 19}\, dx$$
Integral((-4*x + 19)/(3*x^2 + 6*x + 19), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | -4*x + 19 | --------------- dx | 2 | 3*x + 6*x + 19 | /
Reescribimos la función subintegral
3*2*x + 6
2*--------------- /23\
2 |--|
-4*x + 19 3*x + 6*x + 19 \16/
--------------- = - ----------------- + ------------------------
2 3 2
3*x + 6*x + 19 / ___ ___\
|-\/ 3 \/ 3 |
|-------*x - -----| + 1
\ 4 4 / o
/ | | -4*x + 19 | --------------- dx | 2 = | 3*x + 6*x + 19 | /
/
|
| 1
/ 23* | ------------------------ dx
| | 2
| 3*2*x + 6 | / ___ ___\
2* | --------------- dx | |-\/ 3 \/ 3 |
| 2 | |-------*x - -----| + 1
| 3*x + 6*x + 19 | \ 4 4 /
| |
/ /
- ----------------------- + ---------------------------------
3 16 En integral
/
|
| 3*2*x + 6
-2* | --------------- dx
| 2
| 3*x + 6*x + 19
|
/
------------------------
3 hacemos el cambio
2 u = 3*x + 6*x
entonces
integral =
/
|
| 1
-2* | ------ du
| 19 + u
|
/ -2*log(19 + u)
--------------- = --------------
3 3 hacemos cambio inverso
/
|
| 3*2*x + 6
-2* | --------------- dx
| 2
| 3*x + 6*x + 19
| / 2 \
/ -2*log\19 + 3*x + 6*x/
------------------------ = -----------------------
3 3 En integral
/
|
| 1
23* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1
| \ 4 4 /
|
/
---------------------------------
16 hacemos el cambio
___ ___
\/ 3 x*\/ 3
v = - ----- - -------
4 4 entonces
integral =
/
|
| 1
23* | ------ dv
| 2
| 1 + v
|
/ 23*atan(v)
--------------- = ----------
16 16 hacemos cambio inverso
/
|
| 1
23* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 3 \/ 3 |
| |-------*x - -----| + 1 / ___ ___\
| \ 4 4 / ___ |\/ 3 x*\/ 3 |
| 23*\/ 3 *atan|----- + -------|
/ \ 4 4 /
--------------------------------- = ------------------------------
16 12 La solución:
/ ___ ___\
/19 2 \ ___ |\/ 3 x*\/ 3 |
2*log|-- + x + 2*x| 23*\/ 3 *atan|----- + -------|
\3 / \ 4 4 /
C - -------------------- + ------------------------------
3 12
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 3 *(1 + x)| | / 2 \ 23*\/ 3 *atan|-------------| | -4*x + 19 2*log\19 + 3*x + 6*x/ \ 4 / | --------------- dx = C - ---------------------- + ---------------------------- | 2 3 12 | 3*x + 6*x + 19 | /
$$\int \frac{19 - 4 x}{\left(3 x^{2} + 6 x\right) + 19}\, dx = C - \frac{2 \log{\left(3 x^{2} + 6 x + 19 \right)}}{3} + \frac{23 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 1\right)}{4} \right)}}{12}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 3 | ___ |\/ 3 |
23*\/ 3 *atan|-----| 23*\/ 3 *atan|-----|
2*log(28/3) 2*log(19/3) \ 4 / \ 2 /
- ----------- + ----------- - -------------------- + --------------------
3 3 12 12
$$- \frac{2 \log{\left(\frac{28}{3} \right)}}{3} - \frac{23 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{4} \right)}}{12} + \frac{2 \log{\left(\frac{19}{3} \right)}}{3} + \frac{23 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)}}{12}$$
=
=
/ ___\ / ___\
___ |\/ 3 | ___ |\/ 3 |
23*\/ 3 *atan|-----| 23*\/ 3 *atan|-----|
2*log(28/3) 2*log(19/3) \ 4 / \ 2 /
- ----------- + ----------- - -------------------- + --------------------
3 3 12 12
$$- \frac{2 \log{\left(\frac{28}{3} \right)}}{3} - \frac{23 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{4} \right)}}{12} + \frac{2 \log{\left(\frac{19}{3} \right)}}{3} + \frac{23 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3}}{2} \right)}}{12}$$
-2*log(28/3)/3 + 2*log(19/3)/3 - 23*sqrt(3)*atan(sqrt(3)/4)/12 + 23*sqrt(3)*atan(sqrt(3)/2)/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.