Sr Examen
Integral de 1/(x^2+4*x+100) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | x + 4*x + 100 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 100}\, dx$$
Integral(1/(x^2 + 4*x + 100), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | x + 4*x + 100 | /
Reescribimos la función subintegral
1 1
-------------- = -----------------------------
2 / 2 \
x + 4*x + 100 |/ ___ ___\ |
||-\/ 6 \/ 6 | |
96*||-------*x - -----| + 1|
\\ 24 12 / /o
/ | | 1 | -------------- dx | 2 = | x + 4*x + 100 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 24 12 /
|
/
------------------------------
96 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 24 12 /
|
/
------------------------------
96 hacemos el cambio
___ ___
\/ 6 x*\/ 6
v = - ----- - -------
12 24 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
96 96 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1 / ___ ___\
| \ 24 12 / ___ |\/ 6 x*\/ 6 |
| \/ 6 *atan|----- + -------|
/ \ 12 24 /
------------------------------ = ---------------------------
96 24 La solución:
/ ___ ___\
___ |\/ 6 x*\/ 6 |
\/ 6 *atan|----- + -------|
\ 12 24 /
C + ---------------------------
24
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 6 *(2 + x)| | \/ 6 *atan|-------------| | 1 \ 24 / | -------------- dx = C + ------------------------- | 2 24 | x + 4*x + 100 | /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 100}\, dx = C + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6} \left(x + 2\right)}{24} \right)}}{24}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 12 / \ 8 /
- ----------------- + -----------------
24 24
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}}{24} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{8} \right)}}{24}$$
=
=
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 12 / \ 8 /
- ----------------- + -----------------
24 24
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}}{24} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{8} \right)}}{24}$$
-sqrt(6)*atan(sqrt(6)/12)/24 + sqrt(6)*atan(sqrt(6)/8)/24
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.