Sr Examen
Integral de 1/(x^2+4*x+12) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x + 4*x + 12 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 12}\, dx$$
Integral(1/(x^2 + 4*x + 12), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x + 4*x + 12 | /
Reescribimos la función subintegral
1 1
------------- = ----------------------------
2 / 2 \
x + 4*x + 12 |/ ___ ___\ |
||-\/ 2 \/ 2 | |
8*||-------*x - -----| + 1|
\\ 4 2 / /o
/ | | 1 | ------------- dx | 2 = | x + 4*x + 12 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x - -----| + 1
| \ 4 2 /
|
/
------------------------------
8 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x - -----| + 1
| \ 4 2 /
|
/
------------------------------
8 hacemos el cambio
___ ___
\/ 2 x*\/ 2
v = - ----- - -------
2 4 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
8 8 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 2 \/ 2 |
| |-------*x - -----| + 1 / ___ ___\
| \ 4 2 / ___ |\/ 2 x*\/ 2 |
| \/ 2 *atan|----- + -------|
/ \ 2 4 /
------------------------------ = ---------------------------
8 4 La solución:
/ ___ ___\
___ |\/ 2 x*\/ 2 |
\/ 2 *atan|----- + -------|
\ 2 4 /
C + ---------------------------
4
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 2 *(2 + x)| | \/ 2 *atan|-------------| | 1 \ 4 / | ------------- dx = C + ------------------------- | 2 4 | x + 4*x + 12 | /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 12}\, dx = C + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2} \left(x + 2\right)}{4} \right)}}{4}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 2 | ___ |3*\/ 2 |
\/ 2 *atan|-----| \/ 2 *atan|-------|
\ 2 / \ 4 /
- ----------------- + -------------------
4 4
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{4} \right)}}{4}$$
=
=
/ ___\ / ___\
___ |\/ 2 | ___ |3*\/ 2 |
\/ 2 *atan|-----| \/ 2 *atan|-------|
\ 2 / \ 4 /
- ----------------- + -------------------
4 4
$$- \frac{\sqrt{2} \operatorname{atan}{\left(\frac{\sqrt{2}}{2} \right)}}{4} + \frac{\sqrt{2} \operatorname{atan}{\left(\frac{3 \sqrt{2}}{4} \right)}}{4}$$
-sqrt(2)*atan(sqrt(2)/2)/4 + sqrt(2)*atan(3*sqrt(2)/4)/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.