Sr Examen
Integral de 1/(4x^2-5x+4) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 4*x - 5*x + 4 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(4 x^{2} - 5 x\right) + 4}\, dx$$
Integral(1/(4*x^2 - 5*x + 4), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 4*x - 5*x + 4 | /
Reescribimos la función subintegral
1 1
-------------- = ----------------------------------
2 / 2 \
4*x - 5*x + 4 |/ ____ ____\ |
39 ||-8*\/ 39 5*\/ 39 | |
--*||---------*x + --------| + 1|
16 \\ 39 39 / /o
/ | | 1 | -------------- dx | 2 = | 4*x - 5*x + 4 | /
/
|
| 1
16* | ----------------------------- dx
| 2
| / ____ ____\
| |-8*\/ 39 5*\/ 39 |
| |---------*x + --------| + 1
| \ 39 39 /
|
/
--------------------------------------
39 En integral
/
|
| 1
16* | ----------------------------- dx
| 2
| / ____ ____\
| |-8*\/ 39 5*\/ 39 |
| |---------*x + --------| + 1
| \ 39 39 /
|
/
--------------------------------------
39 hacemos el cambio
____ ____
5*\/ 39 8*x*\/ 39
v = -------- - ----------
39 39 entonces
integral =
/
|
| 1
16* | ------ dv
| 2
| 1 + v
|
/ 16*atan(v)
--------------- = ----------
39 39 hacemos cambio inverso
/
|
| 1
16* | ----------------------------- dx
| 2
| / ____ ____\
| |-8*\/ 39 5*\/ 39 |
| |---------*x + --------| + 1 / ____ ____\
| \ 39 39 / ____ | 5*\/ 39 8*x*\/ 39 |
| 2*\/ 39 *atan|- -------- + ----------|
/ \ 39 39 /
-------------------------------------- = --------------------------------------
39 39 La solución:
/ ____ ____\
____ | 5*\/ 39 8*x*\/ 39 |
2*\/ 39 *atan|- -------- + ----------|
\ 39 39 /
C + --------------------------------------
39
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |8*\/ 39 *(-5/8 + x)| | 2*\/ 39 *atan|-------------------| | 1 \ 39 / | -------------- dx = C + ---------------------------------- | 2 39 | 4*x - 5*x + 4 | /
$$\int \frac{1}{\left(4 x^{2} - 5 x\right) + 4}\, dx = C + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{8 \sqrt{39} \left(x - \frac{5}{8}\right)}{39} \right)}}{39}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 39 | ____ |5*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 39 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{5 \sqrt{39}}{39} \right)}}{39}$$
=
=
/ ____\ / ____\
____ |\/ 39 | ____ |5*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 13 / \ 39 /
--------------------- + -----------------------
39 39
$$\frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{13} \right)}}{39} + \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{5 \sqrt{39}}{39} \right)}}{39}$$
2*sqrt(39)*atan(sqrt(39)/13)/39 + 2*sqrt(39)*atan(5*sqrt(39)/39)/39
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.