Sr Examen
Integral de (2x-1)/(5x^2-x+2) dx
Solución
Ha introducido
[src]
1 / | | 2*x - 1 | ------------ dx | 2 | 5*x - x + 2 | / 0
$$\int\limits_{0}^{1} \frac{2 x - 1}{\left(5 x^{2} - x\right) + 2}\, dx$$
Integral((2*x - 1)/(5*x^2 - x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x - 1 | ------------ dx | 2 | 5*x - x + 2 | /
Reescribimos la función subintegral
/-4 \
/ 5*2*x - 1 \ |----|
|------------| | 39|
| 2 | |5*--|
2*x - 1 \5*x - x + 2/ \ 20/
------------ = -------------- + ----------------------------
2 5 2
5*x - x + 2 / ____ ____\
|-10*\/ 39 \/ 39 |
|----------*x + ------| + 1
\ 39 39 / o
/ | | 2*x - 1 | ------------ dx | 2 = | 5*x - x + 2 | /
/
|
| 1
16* | ---------------------------- dx /
| 2 |
| / ____ ____\ | 5*2*x - 1
| |-10*\/ 39 \/ 39 | | ------------ dx
| |----------*x + ------| + 1 | 2
| \ 39 39 / | 5*x - x + 2
| |
/ /
- ------------------------------------- + ------------------
39 5 En integral
/
|
| 5*2*x - 1
| ------------ dx
| 2
| 5*x - x + 2
|
/
------------------
5 hacemos el cambio
2 u = -x + 5*x
entonces
integral =
/
|
| 1
| ----- du
| 2 + u
|
/ log(2 + u)
----------- = ----------
5 5 hacemos cambio inverso
/
|
| 5*2*x - 1
| ------------ dx
| 2
| 5*x - x + 2
| / 2\
/ log\2 - x + 5*x /
------------------ = -----------------
5 5 En integral
/
|
| 1
-16* | ---------------------------- dx
| 2
| / ____ ____\
| |-10*\/ 39 \/ 39 |
| |----------*x + ------| + 1
| \ 39 39 /
|
/
--------------------------------------
39 hacemos el cambio
____ ____
\/ 39 10*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
| / ____ ____\
| |-10*\/ 39 \/ 39 |
| |----------*x + ------| + 1 / ____ ____\
| \ 39 39 / ____ | \/ 39 10*x*\/ 39 |
| -8*\/ 39 *atan|- ------ + -----------|
/ \ 39 39 /
-------------------------------------- = --------------------------------------
39 195 La solución:
/ ____ ____\
/2 2 x\ ____ | \/ 39 10*x*\/ 39 |
log|- + x - -| 8*\/ 39 *atan|- ------ + -----------|
\5 5/ \ 39 39 /
C + --------------- - -------------------------------------
5 195
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |10*\/ 39 *(-1/10 + x)| | / 2\ 8*\/ 39 *atan|---------------------| | 2*x - 1 log\2 - x + 5*x / \ 39 / | ------------ dx = C + ----------------- - ------------------------------------ | 2 5 195 | 5*x - x + 2 | /
$$\int \frac{2 x - 1}{\left(5 x^{2} - x\right) + 2}\, dx = C + \frac{\log{\left(5 x^{2} - x + 2 \right)}}{5} - \frac{8 \sqrt{39} \operatorname{atan}{\left(\frac{10 \sqrt{39} \left(x - \frac{1}{10}\right)}{39} \right)}}{195}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
8*\/ 39 *atan|------| 8*\/ 39 *atan|--------|
log(2/5) log(6/5) \ 39 / \ 13 /
- -------- + -------- - --------------------- - -----------------------
5 5 195 195
$$- \frac{8 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{195} - \frac{8 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{195} + \frac{\log{\left(\frac{6}{5} \right)}}{5} - \frac{\log{\left(\frac{2}{5} \right)}}{5}$$
=
=
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
8*\/ 39 *atan|------| 8*\/ 39 *atan|--------|
log(2/5) log(6/5) \ 39 / \ 13 /
- -------- + -------- - --------------------- - -----------------------
5 5 195 195
$$- \frac{8 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{195} - \frac{8 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{195} + \frac{\log{\left(\frac{6}{5} \right)}}{5} - \frac{\log{\left(\frac{2}{5} \right)}}{5}$$
-log(2/5)/5 + log(6/5)/5 - 8*sqrt(39)*atan(sqrt(39)/39)/195 - 8*sqrt(39)*atan(3*sqrt(39)/13)/195
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.