Sr Examen
Integral de 1/(x^2-4*x+10) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x - 4*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - 4 x\right) + 10}\, dx$$
Integral(1/(x^2 - 4*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x - 4*x + 10 | /
Reescribimos la función subintegral
1 1
------------- = ----------------------------
2 / 2 \
x - 4*x + 10 |/ ___ ___\ |
||-\/ 6 \/ 6 | |
6*||-------*x + -----| + 1|
\\ 6 3 / /o
/ | | 1 | ------------- dx | 2 = | x - 4*x + 10 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x + -----| + 1
| \ 6 3 /
|
/
------------------------------
6 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x + -----| + 1
| \ 6 3 /
|
/
------------------------------
6 hacemos el cambio
___ ___
\/ 6 x*\/ 6
v = ----- - -------
3 6 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
6 6 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x + -----| + 1 / ___ ___\
| \ 6 3 / ___ | \/ 6 x*\/ 6 |
| \/ 6 *atan|- ----- + -------|
/ \ 3 6 /
------------------------------ = -----------------------------
6 6 La solución:
/ ___ ___\
___ | \/ 6 x*\/ 6 |
\/ 6 *atan|- ----- + -------|
\ 3 6 /
C + -----------------------------
6
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 6 *(-2 + x)| | \/ 6 *atan|--------------| | 1 \ 6 / | ------------- dx = C + -------------------------- | 2 6 | x - 4*x + 10 | /
$$\int \frac{1}{\left(x^{2} - 4 x\right) + 10}\, dx = C + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6} \left(x - 2\right)}{6} \right)}}{6}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 6 / \ 3 /
- ----------------- + -----------------
6 6
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{6} \right)}}{6} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{6}$$
=
=
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
\/ 6 *atan|-----| \/ 6 *atan|-----|
\ 6 / \ 3 /
- ----------------- + -----------------
6 6
$$- \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{6} \right)}}{6} + \frac{\sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{6}$$
-sqrt(6)*atan(sqrt(6)/6)/6 + sqrt(6)*atan(sqrt(6)/3)/6
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.