Sr Examen
Integral de 1/(x²+10x+30) dx
Solución
Ha introducido
[src]
1 / | | 1 | -------------- dx | 2 | x + 10*x + 30 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{2} + 10 x\right) + 30}\, dx$$
Integral(1/(x^2 + 10*x + 30), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 1 | -------------- dx | 2 | x + 10*x + 30 | /
Reescribimos la función subintegral
1 1
-------------- = ----------------------------
2 / 2 \
x + 10*x + 30 |/ ___ \ |
||-\/ 5 ___| |
5*||-------*x - \/ 5 | + 1|
\\ 5 / /o
/ | | 1 | -------------- dx | 2 = | x + 10*x + 30 | /
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 5 ___|
| |-------*x - \/ 5 | + 1
| \ 5 /
|
/
------------------------------
5 En integral
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 5 ___|
| |-------*x - \/ 5 | + 1
| \ 5 /
|
/
------------------------------
5 hacemos el cambio
___
___ x*\/ 5
v = - \/ 5 - -------
5 entonces
integral =
/
|
| 1
| ------ dv
| 2
| 1 + v
|
/ atan(v)
------------ = -------
5 5 hacemos cambio inverso
/
|
| 1
| ------------------------ dx
| 2
| / ___ \
| |-\/ 5 ___|
| |-------*x - \/ 5 | + 1 / ___\
| \ 5 / ___ | ___ x*\/ 5 |
| \/ 5 *atan|\/ 5 + -------|
/ \ 5 /
------------------------------ = ---------------------------
5 5 La solución:
/ ___\
___ | ___ x*\/ 5 |
\/ 5 *atan|\/ 5 + -------|
\ 5 /
C + ---------------------------
5
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 5 *(5 + x)| | \/ 5 *atan|-------------| | 1 \ 5 / | -------------- dx = C + ------------------------- | 2 5 | x + 10*x + 30 | /
$$\int \frac{1}{\left(x^{2} + 10 x\right) + 30}\, dx = C + \frac{\sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(x + 5\right)}{5} \right)}}{5}$$
Gráfica
Respuesta
[src]
/ ___\
___ |6*\/ 5 |
___ / ___\ \/ 5 *atan|-------|
\/ 5 *atan\\/ 5 / \ 5 /
- ----------------- + -------------------
5 5
$$- \frac{\sqrt{5} \operatorname{atan}{\left(\sqrt{5} \right)}}{5} + \frac{\sqrt{5} \operatorname{atan}{\left(\frac{6 \sqrt{5}}{5} \right)}}{5}$$
=
=
/ ___\
___ |6*\/ 5 |
___ / ___\ \/ 5 *atan|-------|
\/ 5 *atan\\/ 5 / \ 5 /
- ----------------- + -------------------
5 5
$$- \frac{\sqrt{5} \operatorname{atan}{\left(\sqrt{5} \right)}}{5} + \frac{\sqrt{5} \operatorname{atan}{\left(\frac{6 \sqrt{5}}{5} \right)}}{5}$$
-sqrt(5)*atan(sqrt(5))/5 + sqrt(5)*atan(6*sqrt(5)/5)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.