Sr Examen
Integral de dx/(2*x^2+8*x+20) dx
Solución
Ha introducido
[src]
1 / | | 1 | --------------- dx | 2 | 2*x + 8*x + 20 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(2 x^{2} + 8 x\right) + 20}\, dx$$
Integral(1/(2*x^2 + 8*x + 20), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | --------------- dx | 2 | 2*x + 8*x + 20 | /
Reescribimos la función subintegral
1 1
--------------- = -----------------------------
2 / 2 \
2*x + 8*x + 20 |/ ___ ___\ |
||-\/ 6 \/ 6 | |
12*||-------*x - -----| + 1|
\\ 6 3 / /o
/ | | 1 | --------------- dx | 2 = | 2*x + 8*x + 20 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 6 3 /
|
/
------------------------------
12 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 6 3 /
|
/
------------------------------
12 hacemos el cambio
___ ___
\/ 6 x*\/ 6
v = - ----- - -------
3 6 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
12 12 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1 / ___ ___\
| \ 6 3 / ___ |\/ 6 x*\/ 6 |
| \/ 6 *atan|----- + -------|
/ \ 3 6 /
------------------------------ = ---------------------------
12 12 La solución:
/ ___ ___\
___ |\/ 6 x*\/ 6 |
\/ 6 *atan|----- + -------|
\ 3 6 /
C + ---------------------------
12
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 6 *(2 + x)| | \/ 6 *atan|-------------| | 1 \ 6 / | --------------- dx = C + ------------------------- | 2 12 | 2*x + 8*x + 20 | /
$$\int \frac{1}{\left(2 x^{2} + 8 x\right) + 20}\, dx = C + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6} \left(x + 2\right)}{6} \right)}}{12}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 3 / \ 2 /
- ----------------- + -----------------
12 12
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{12} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{12}$$
=
=
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 3 / \ 2 /
- ----------------- + -----------------
12 12
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{12} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{12}$$
-sqrt(6)*atan(sqrt(6)/3)/12 + sqrt(6)*atan(sqrt(6)/2)/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.