Sr Examen
Integral de (3*x-2)/((2-3*x+5*x^2)) dx
Solución
Ha introducido
[src]
1 / | | 3*x - 2 | -------------- dx | 2 | 2 - 3*x + 5*x | / 0
$$\int\limits_{0}^{1} \frac{3 x - 2}{5 x^{2} + \left(2 - 3 x\right)}\, dx$$
Integral((3*x - 2)/(2 - 3*x + 5*x^2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x - 2 | -------------- dx | 2 | 2 - 3*x + 5*x | /
Reescribimos la función subintegral
/ -11 \
5*2*x - 3 |-----|
3*-------------- | 31|
2 |10*--|
3*x - 2 5*x - 3*x + 2 \ 20/
-------------- = ---------------- + ------------------------------
2 10 2
2 - 3*x + 5*x / ____ ____\
|-10*\/ 31 3*\/ 31 |
|----------*x + --------| + 1
\ 31 31 / o
/ | | 3*x - 2 | -------------- dx | 2 = | 2 - 3*x + 5*x | /
/
|
| 1
22* | ------------------------------ dx /
| 2 |
| / ____ ____\ | 5*2*x - 3
| |-10*\/ 31 3*\/ 31 | 3* | -------------- dx
| |----------*x + --------| + 1 | 2
| \ 31 31 / | 5*x - 3*x + 2
| |
/ /
- --------------------------------------- + ----------------------
31 10 En integral
/
|
| 5*2*x - 3
3* | -------------- dx
| 2
| 5*x - 3*x + 2
|
/
----------------------
10 hacemos el cambio
2 u = -3*x + 5*x
entonces
integral =
/
|
| 1
3* | ----- du
| 2 + u
|
/ 3*log(2 + u)
------------- = ------------
10 10 hacemos cambio inverso
/
|
| 5*2*x - 3
3* | -------------- dx
| 2
| 5*x - 3*x + 2
| / 2\
/ 3*log\2 - 3*x + 5*x /
---------------------- = ---------------------
10 10 En integral
/
|
| 1
-22* | ------------------------------ dx
| 2
| / ____ ____\
| |-10*\/ 31 3*\/ 31 |
| |----------*x + --------| + 1
| \ 31 31 /
|
/
----------------------------------------
31 hacemos el cambio
____ ____
3*\/ 31 10*x*\/ 31
v = -------- - -----------
31 31 entonces
integral =
/
|
| 1
-22* | ------ dv
| 2
| 1 + v
|
/ -22*atan(v)
---------------- = -----------
31 31 hacemos cambio inverso
/
|
| 1
-22* | ------------------------------ dx
| 2
| / ____ ____\
| |-10*\/ 31 3*\/ 31 |
| |----------*x + --------| + 1 / ____ ____\
| \ 31 31 / ____ | 3*\/ 31 10*x*\/ 31 |
| -11*\/ 31 *atan|- -------- + -----------|
/ \ 31 31 /
---------------------------------------- = -----------------------------------------
31 155 La solución:
/ ____ ____\
/2 2 3*x\ ____ | 3*\/ 31 10*x*\/ 31 |
3*log|- + x - ---| 11*\/ 31 *atan|- -------- + -----------|
\5 5 / \ 31 31 /
C + ------------------- - ----------------------------------------
10 155
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |10*\/ 31 *(-3/10 + x)| | / 2\ 11*\/ 31 *atan|---------------------| | 3*x - 2 3*log\2 - 3*x + 5*x / \ 31 / | -------------- dx = C + --------------------- - ------------------------------------- | 2 10 155 | 2 - 3*x + 5*x | /
$$\int \frac{3 x - 2}{5 x^{2} + \left(2 - 3 x\right)}\, dx = C + \frac{3 \log{\left(5 x^{2} - 3 x + 2 \right)}}{10} - \frac{11 \sqrt{31} \operatorname{atan}{\left(\frac{10 \sqrt{31} \left(x - \frac{3}{10}\right)}{31} \right)}}{155}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |3*\/ 31 | ____ |7*\/ 31 |
11*\/ 31 *atan|--------| 11*\/ 31 *atan|--------|
3*log(2/5) 3*log(4/5) \ 31 / \ 31 /
- ---------- + ---------- - ------------------------ - ------------------------
10 10 155 155
$$- \frac{11 \sqrt{31} \operatorname{atan}{\left(\frac{7 \sqrt{31}}{31} \right)}}{155} - \frac{11 \sqrt{31} \operatorname{atan}{\left(\frac{3 \sqrt{31}}{31} \right)}}{155} + \frac{3 \log{\left(\frac{4}{5} \right)}}{10} - \frac{3 \log{\left(\frac{2}{5} \right)}}{10}$$
=
=
/ ____\ / ____\
____ |3*\/ 31 | ____ |7*\/ 31 |
11*\/ 31 *atan|--------| 11*\/ 31 *atan|--------|
3*log(2/5) 3*log(4/5) \ 31 / \ 31 /
- ---------- + ---------- - ------------------------ - ------------------------
10 10 155 155
$$- \frac{11 \sqrt{31} \operatorname{atan}{\left(\frac{7 \sqrt{31}}{31} \right)}}{155} - \frac{11 \sqrt{31} \operatorname{atan}{\left(\frac{3 \sqrt{31}}{31} \right)}}{155} + \frac{3 \log{\left(\frac{4}{5} \right)}}{10} - \frac{3 \log{\left(\frac{2}{5} \right)}}{10}$$
-3*log(2/5)/10 + 3*log(4/5)/10 - 11*sqrt(31)*atan(3*sqrt(31)/31)/155 - 11*sqrt(31)*atan(7*sqrt(31)/31)/155
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.