Sr Examen
Integral de 1/(x^2-3*x+5) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------ dx | 2 | x - 3*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} - 3 x\right) + 5}\, dx$$
Integral(1/(x^2 - 3*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------ dx | 2 | x - 3*x + 5 | /
Reescribimos la función subintegral
1 1
------------ = ------------------------------------
2 / 2 \
x - 3*x + 5 |/ ____ ____\ |
||-2*\/ 11 3*\/ 11 | |
11/4*||---------*x + --------| + 1|
\\ 11 11 / /o
/ | | 1 | ------------ dx | 2 = | x - 3*x + 5 | /
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 11 3*\/ 11 |
| |---------*x + --------| + 1
| \ 11 11 /
|
/
-------------------------------------
11 En integral
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 11 3*\/ 11 |
| |---------*x + --------| + 1
| \ 11 11 /
|
/
-------------------------------------
11 hacemos el cambio
____ ____
3*\/ 11 2*x*\/ 11
v = -------- - ----------
11 11 entonces
integral =
/
|
| 1
4* | ------ dv
| 2
| 1 + v
|
/ 4*atan(v)
-------------- = ---------
11 11 hacemos cambio inverso
/
|
| 1
4* | ----------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 11 3*\/ 11 |
| |---------*x + --------| + 1 / ____ ____\
| \ 11 11 / ____ | 3*\/ 11 2*x*\/ 11 |
| 2*\/ 11 *atan|- -------- + ----------|
/ \ 11 11 /
------------------------------------- = --------------------------------------
11 11 La solución:
/ ____ ____\
____ | 3*\/ 11 2*x*\/ 11 |
2*\/ 11 *atan|- -------- + ----------|
\ 11 11 /
C + --------------------------------------
11
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 11 *(-3/2 + x)| | 2*\/ 11 *atan|-------------------| | 1 \ 11 / | ------------ dx = C + ---------------------------------- | 2 11 | x - 3*x + 5 | /
$$\int \frac{1}{\left(x^{2} - 3 x\right) + 5}\, dx = C + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{2 \sqrt{11} \left(x - \frac{3}{2}\right)}{11} \right)}}{11}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 11 | ____ |3*\/ 11 |
2*\/ 11 *atan|------| 2*\/ 11 *atan|--------|
\ 11 / \ 11 /
- --------------------- + -----------------------
11 11
$$- \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{3 \sqrt{11}}{11} \right)}}{11}$$
=
=
/ ____\ / ____\
____ |\/ 11 | ____ |3*\/ 11 |
2*\/ 11 *atan|------| 2*\/ 11 *atan|--------|
\ 11 / \ 11 /
- --------------------- + -----------------------
11 11
$$- \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{\sqrt{11}}{11} \right)}}{11} + \frac{2 \sqrt{11} \operatorname{atan}{\left(\frac{3 \sqrt{11}}{11} \right)}}{11}$$
-2*sqrt(11)*atan(sqrt(11)/11)/11 + 2*sqrt(11)*atan(3*sqrt(11)/11)/11
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.