Sr Examen
Integral de 1/(7*x^2-2*x+5) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | 7*x - 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(7 x^{2} - 2 x\right) + 5}\, dx$$
Integral(1/(7*x^2 - 2*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | 7*x - 2*x + 5 | /
Reescribimos la función subintegral
1 1
-------------- = ----------------------------------
2 / 2 \
7*x - 2*x + 5 |/ ____ ____\ |
||-7*\/ 34 \/ 34 | |
34/7*||---------*x + ------| + 1|
\\ 34 34 / /o
/ | | 1 | -------------- dx | 2 = | 7*x - 2*x + 5 | /
/
|
| 1
7* | --------------------------- dx
| 2
| / ____ ____\
| |-7*\/ 34 \/ 34 |
| |---------*x + ------| + 1
| \ 34 34 /
|
/
-----------------------------------
34 En integral
/
|
| 1
7* | --------------------------- dx
| 2
| / ____ ____\
| |-7*\/ 34 \/ 34 |
| |---------*x + ------| + 1
| \ 34 34 /
|
/
-----------------------------------
34 hacemos el cambio
____ ____
\/ 34 7*x*\/ 34
v = ------ - ----------
34 34 entonces
integral =
/
|
| 1
7* | ------ dv
| 2
| 1 + v
|
/ 7*atan(v)
-------------- = ---------
34 34 hacemos cambio inverso
/
|
| 1
7* | --------------------------- dx
| 2
| / ____ ____\
| |-7*\/ 34 \/ 34 |
| |---------*x + ------| + 1 / ____ ____\
| \ 34 34 / ____ | \/ 34 7*x*\/ 34 |
| \/ 34 *atan|- ------ + ----------|
/ \ 34 34 /
----------------------------------- = ----------------------------------
34 34 La solución:
/ ____ ____\
____ | \/ 34 7*x*\/ 34 |
\/ 34 *atan|- ------ + ----------|
\ 34 34 /
C + ----------------------------------
34
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |7*\/ 34 *(-1/7 + x)| | \/ 34 *atan|-------------------| | 1 \ 34 / | -------------- dx = C + -------------------------------- | 2 34 | 7*x - 2*x + 5 | /
$$\int \frac{1}{\left(7 x^{2} - 2 x\right) + 5}\, dx = C + \frac{\sqrt{34} \operatorname{atan}{\left(\frac{7 \sqrt{34} \left(x - \frac{1}{7}\right)}{34} \right)}}{34}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 34 | ____ |3*\/ 34 |
\/ 34 *atan|------| \/ 34 *atan|--------|
\ 34 / \ 17 /
------------------- + ---------------------
34 34
$$\frac{\sqrt{34} \operatorname{atan}{\left(\frac{\sqrt{34}}{34} \right)}}{34} + \frac{\sqrt{34} \operatorname{atan}{\left(\frac{3 \sqrt{34}}{17} \right)}}{34}$$
=
=
/ ____\ / ____\
____ |\/ 34 | ____ |3*\/ 34 |
\/ 34 *atan|------| \/ 34 *atan|--------|
\ 34 / \ 17 /
------------------- + ---------------------
34 34
$$\frac{\sqrt{34} \operatorname{atan}{\left(\frac{\sqrt{34}}{34} \right)}}{34} + \frac{\sqrt{34} \operatorname{atan}{\left(\frac{3 \sqrt{34}}{17} \right)}}{34}$$
sqrt(34)*atan(sqrt(34)/34)/34 + sqrt(34)*atan(3*sqrt(34)/17)/34
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.