Sr Examen
Integral de dx/(3x^2+3x+6) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 3*x + 3*x + 6 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 x^{2} + 3 x\right) + 6}\, dx$$
Integral(1/(3*x^2 + 3*x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 3*x + 3*x + 6 | /
Reescribimos la función subintegral
1 1
-------------- = --------------------------------
2 / 2 \
3*x + 3*x + 6 |/ ___ ___\ |
||-2*\/ 7 \/ 7 | |
21/4*||--------*x - -----| + 1|
\\ 7 7 / /o
/ | | 1 | -------------- dx | 2 = | 3*x + 3*x + 6 | /
/
|
| 1
4* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1
| \ 7 7 /
|
/
---------------------------------
21 En integral
/
|
| 1
4* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1
| \ 7 7 /
|
/
---------------------------------
21 hacemos el cambio
___ ___
\/ 7 2*x*\/ 7
v = - ----- - ---------
7 7 entonces
integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
21 21 hacemos cambio inverso
/
|
| 1
4* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1 / ___ ___\
| \ 7 7 / ___ |\/ 7 2*x*\/ 7 |
| 2*\/ 7 *atan|----- + ---------|
/ \ 7 7 /
--------------------------------- = -------------------------------
21 21 La solución:
/ ___ ___\
___ |\/ 7 2*x*\/ 7 |
2*\/ 7 *atan|----- + ---------|
\ 7 7 /
C + -------------------------------
21
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(1/2 + x)| | 2*\/ 7 *atan|-----------------| | 1 \ 7 / | -------------- dx = C + ------------------------------- | 2 21 | 3*x + 3*x + 6 | /
$$\int \frac{1}{\left(3 x^{2} + 3 x\right) + 6}\, dx = C + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x + \frac{1}{2}\right)}{7} \right)}}{21}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
2*\/ 7 *atan|-----| 2*\/ 7 *atan|-------|
\ 7 / \ 7 /
- ------------------- + ---------------------
21 21
$$- \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{21} + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{21}$$
=
=
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
2*\/ 7 *atan|-----| 2*\/ 7 *atan|-------|
\ 7 / \ 7 /
- ------------------- + ---------------------
21 21
$$- \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{21} + \frac{2 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{21}$$
-2*sqrt(7)*atan(sqrt(7)/7)/21 + 2*sqrt(7)*atan(3*sqrt(7)/7)/21
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.