Sr Examen
Integral de dx/(-5x^2+9x-6) dx
Solución
Ha introducido
[src]
1 / | | 1 | ---------------- dx | 2 | - 5*x + 9*x - 6 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(- 5 x^{2} + 9 x\right) - 6}\, dx$$
Integral(1/(-5*x^2 + 9*x - 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ---------------- dx | 2 | - 5*x + 9*x - 6 | /
Reescribimos la función subintegral
1 1
---------------- = -------------------------------------
2 / 2 \
- 5*x + 9*x - 6 |/ ____ ____\ |
-39 ||-10*\/ 39 3*\/ 39 | |
----*||----------*x + --------| + 1|
20 \\ 39 13 / /o
/ | | 1 | ---------------- dx | 2 = | - 5*x + 9*x - 6 | /
/
|
| 1
-20* | ------------------------------ dx
| 2
| / ____ ____\
| |-10*\/ 39 3*\/ 39 |
| |----------*x + --------| + 1
| \ 39 13 /
|
/
----------------------------------------
39 En integral
/
|
| 1
-20* | ------------------------------ dx
| 2
| / ____ ____\
| |-10*\/ 39 3*\/ 39 |
| |----------*x + --------| + 1
| \ 39 13 /
|
/
----------------------------------------
39 hacemos el cambio
____ ____
3*\/ 39 10*x*\/ 39
v = -------- - -----------
13 39 entonces
integral =
/
|
| 1
-20* | ------ dv
| 2
| 1 + v
|
/ -20*atan(v)
---------------- = -----------
39 39 hacemos cambio inverso
/
|
| 1
-20* | ------------------------------ dx
| 2
| / ____ ____\
| |-10*\/ 39 3*\/ 39 |
| |----------*x + --------| + 1 / ____ ____\
| \ 39 13 / ____ | 3*\/ 39 10*x*\/ 39 |
| -2*\/ 39 *atan|- -------- + -----------|
/ \ 13 39 /
---------------------------------------- = ----------------------------------------
39 39 La solución:
/ ____ ____\
____ | 3*\/ 39 10*x*\/ 39 |
2*\/ 39 *atan|- -------- + -----------|
\ 13 39 /
C - ---------------------------------------
39
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |10*\/ 39 *(-9/10 + x)| | 2*\/ 39 *atan|---------------------| | 1 \ 39 / | ---------------- dx = C - ------------------------------------ | 2 39 | - 5*x + 9*x - 6 | /
$$\int \frac{1}{\left(- 5 x^{2} + 9 x\right) - 6}\, dx = C - \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{10 \sqrt{39} \left(x - \frac{9}{10}\right)}{39} \right)}}{39}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 39 / \ 13 /
- --------------------- - -----------------------
39 39
$$- \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39} - \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{39}$$
=
=
/ ____\ / ____\
____ |\/ 39 | ____ |3*\/ 39 |
2*\/ 39 *atan|------| 2*\/ 39 *atan|--------|
\ 39 / \ 13 /
- --------------------- - -----------------------
39 39
$$- \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{3 \sqrt{39}}{13} \right)}}{39} - \frac{2 \sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{39}$$
-2*sqrt(39)*atan(sqrt(39)/39)/39 - 2*sqrt(39)*atan(3*sqrt(39)/13)/39
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.