Sr Examen
Integral de 1/(2x^2-5x+7) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 2*x - 5*x + 7 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(2 x^{2} - 5 x\right) + 7}\, dx$$
Integral(1/(2*x^2 - 5*x + 7), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 2*x - 5*x + 7 | /
Reescribimos la función subintegral
1 1
-------------- = ------------------------------------
2 / 2 \
2*x - 5*x + 7 |/ ____ ____\ |
||-4*\/ 31 5*\/ 31 | |
31/8*||---------*x + --------| + 1|
\\ 31 31 / /o
/ | | 1 | -------------- dx | 2 = | 2*x - 5*x + 7 | /
/
|
| 1
8* | ----------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 31 5*\/ 31 |
| |---------*x + --------| + 1
| \ 31 31 /
|
/
-------------------------------------
31 En integral
/
|
| 1
8* | ----------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 31 5*\/ 31 |
| |---------*x + --------| + 1
| \ 31 31 /
|
/
-------------------------------------
31 hacemos el cambio
____ ____
5*\/ 31 4*x*\/ 31
v = -------- - ----------
31 31 entonces
integral =
/
|
| 1
8* | ------ dv
| 2
| 1 + v
|
/ 8*atan(v)
-------------- = ---------
31 31 hacemos cambio inverso
/
|
| 1
8* | ----------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 31 5*\/ 31 |
| |---------*x + --------| + 1 / ____ ____\
| \ 31 31 / ____ | 5*\/ 31 4*x*\/ 31 |
| 2*\/ 31 *atan|- -------- + ----------|
/ \ 31 31 /
------------------------------------- = --------------------------------------
31 31 La solución:
/ ____ ____\
____ | 5*\/ 31 4*x*\/ 31 |
2*\/ 31 *atan|- -------- + ----------|
\ 31 31 /
C + --------------------------------------
31
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |4*\/ 31 *(-5/4 + x)| | 2*\/ 31 *atan|-------------------| | 1 \ 31 / | -------------- dx = C + ---------------------------------- | 2 31 | 2*x - 5*x + 7 | /
$$\int \frac{1}{\left(2 x^{2} - 5 x\right) + 7}\, dx = C + \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{4 \sqrt{31} \left(x - \frac{5}{4}\right)}{31} \right)}}{31}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 31 | ____ |5*\/ 31 |
2*\/ 31 *atan|------| 2*\/ 31 *atan|--------|
\ 31 / \ 31 /
- --------------------- + -----------------------
31 31
$$- \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31}}{31} \right)}}{31} + \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{5 \sqrt{31}}{31} \right)}}{31}$$
=
=
/ ____\ / ____\
____ |\/ 31 | ____ |5*\/ 31 |
2*\/ 31 *atan|------| 2*\/ 31 *atan|--------|
\ 31 / \ 31 /
- --------------------- + -----------------------
31 31
$$- \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{\sqrt{31}}{31} \right)}}{31} + \frac{2 \sqrt{31} \operatorname{atan}{\left(\frac{5 \sqrt{31}}{31} \right)}}{31}$$
-2*sqrt(31)*atan(sqrt(31)/31)/31 + 2*sqrt(31)*atan(5*sqrt(31)/31)/31
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.