Sr Examen
Integral de dx/(x^2+4x+25) dx
Solución
Ha introducido
[src]
1 / | | 1 | ------------- dx | 2 | x + 4*x + 25 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 4 x\right) + 25}\, dx$$
Integral(1/(x^2 + 4*x + 25), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | ------------- dx | 2 | x + 4*x + 25 | /
Reescribimos la función subintegral
1 1
------------- = ---------------------------------
2 / 2 \
x + 4*x + 25 |/ ____ ____\ |
||-\/ 21 2*\/ 21 | |
21*||--------*x - --------| + 1|
\\ 21 21 / /o
/ | | 1 | ------------- dx | 2 = | x + 4*x + 25 | /
/
|
| 1
| ---------------------------- dx
| 2
| / ____ ____\
| |-\/ 21 2*\/ 21 |
| |--------*x - --------| + 1
| \ 21 21 /
|
/
----------------------------------
21 En integral
/
|
| 1
| ---------------------------- dx
| 2
| / ____ ____\
| |-\/ 21 2*\/ 21 |
| |--------*x - --------| + 1
| \ 21 21 /
|
/
----------------------------------
21 hacemos el cambio
____ ____
2*\/ 21 x*\/ 21
v = - -------- - --------
21 21 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
21 21 hacemos cambio inverso
/
|
| 1
| ---------------------------- dx
| 2
| / ____ ____\
| |-\/ 21 2*\/ 21 |
| |--------*x - --------| + 1 / ____ ____\
| \ 21 21 / ____ |2*\/ 21 x*\/ 21 |
| \/ 21 *atan|-------- + --------|
/ \ 21 21 /
---------------------------------- = --------------------------------
21 21 La solución:
/ ____ ____\
____ |2*\/ 21 x*\/ 21 |
\/ 21 *atan|-------- + --------|
\ 21 21 /
C + --------------------------------
21
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |\/ 21 *(2 + x)| | \/ 21 *atan|--------------| | 1 \ 21 / | ------------- dx = C + --------------------------- | 2 21 | x + 4*x + 25 | /
$$\int \frac{1}{\left(x^{2} + 4 x\right) + 25}\, dx = C + \frac{\sqrt{21} \operatorname{atan}{\left(\frac{\sqrt{21} \left(x + 2\right)}{21} \right)}}{21}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |2*\/ 21 | ____ |\/ 21 |
\/ 21 *atan|--------| \/ 21 *atan|------|
\ 21 / \ 7 /
- --------------------- + -------------------
21 21
$$- \frac{\sqrt{21} \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)}}{21} + \frac{\sqrt{21} \operatorname{atan}{\left(\frac{\sqrt{21}}{7} \right)}}{21}$$
=
=
/ ____\ / ____\
____ |2*\/ 21 | ____ |\/ 21 |
\/ 21 *atan|--------| \/ 21 *atan|------|
\ 21 / \ 7 /
- --------------------- + -------------------
21 21
$$- \frac{\sqrt{21} \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)}}{21} + \frac{\sqrt{21} \operatorname{atan}{\left(\frac{\sqrt{21}}{7} \right)}}{21}$$
-sqrt(21)*atan(2*sqrt(21)/21)/21 + sqrt(21)*atan(sqrt(21)/7)/21
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.