Sr Examen
Integral de 2*x/(x^2+4*x+10) dx
Solución
Ha introducido
[src]
1 / | | 2*x | ------------- dx | 2 | x + 4*x + 10 | / 0
$$\int\limits_{0}^{1} \frac{2 x}{\left(x^{2} + 4 x\right) + 10}\, dx$$
Integral((2*x)/(x^2 + 4*x + 10), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x | ------------- dx | 2 | x + 4*x + 10 | /
Reescribimos la función subintegral
/-4 \
|---|
2*x 2*x + 4 \ 6 /
------------- = ------------- + ------------------------
2 2 2
x + 4*x + 10 x + 4*x + 10 / ___ ___\
|-\/ 6 \/ 6 |
|-------*x - -----| + 1
\ 6 3 / o
/ | | 2*x | ------------- dx | 2 = | x + 4*x + 10 | /
/
|
| 1
2* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 6 3 / /
| |
/ | 2*x + 4
- -------------------------------- + | ------------- dx
3 | 2
| x + 4*x + 10
|
/ En integral
/ | | 2*x + 4 | ------------- dx | 2 | x + 4*x + 10 | /
hacemos el cambio
2 u = x + 4*x
entonces
integral =
/ | | 1 | ------ du = log(10 + u) | 10 + u | /
hacemos cambio inverso
/ | | 2*x + 4 / 2 \ | ------------- dx = log\10 + x + 4*x/ | 2 | x + 4*x + 10 | /
En integral
/
|
| 1
-2* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1
| \ 6 3 /
|
/
---------------------------------
3 hacemos el cambio
___ ___
\/ 6 x*\/ 6
v = - ----- - -------
3 6 entonces
integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
3 3 hacemos cambio inverso
/
|
| 1
-2* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 6 \/ 6 |
| |-------*x - -----| + 1 / ___ ___\
| \ 6 3 / ___ |\/ 6 x*\/ 6 |
| -2*\/ 6 *atan|----- + -------|
/ \ 3 6 /
--------------------------------- = ------------------------------
3 3 La solución:
/ ___ ___\
___ |\/ 6 x*\/ 6 |
2*\/ 6 *atan|----- + -------|
\ 3 6 / / 2 \
C - ----------------------------- + log\10 + x + 4*x/
3
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 6 *(2 + x)| | 2*\/ 6 *atan|-------------| | 2*x \ 6 / / 2 \ | ------------- dx = C - --------------------------- + log\10 + x + 4*x/ | 2 3 | x + 4*x + 10 | /
$$\int \frac{2 x}{\left(x^{2} + 4 x\right) + 10}\, dx = C + \log{\left(x^{2} + 4 x + 10 \right)} - \frac{2 \sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6} \left(x + 2\right)}{6} \right)}}{3}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
2*\/ 6 *atan|-----| 2*\/ 6 *atan|-----|
\ 2 / \ 3 /
-log(10) - ------------------- + ------------------- + log(15)
3 3
$$- \log{\left(10 \right)} - \frac{2 \sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{3} + \frac{2 \sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{3} + \log{\left(15 \right)}$$
=
=
/ ___\ / ___\
___ |\/ 6 | ___ |\/ 6 |
2*\/ 6 *atan|-----| 2*\/ 6 *atan|-----|
\ 2 / \ 3 /
-log(10) - ------------------- + ------------------- + log(15)
3 3
$$- \log{\left(10 \right)} - \frac{2 \sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{2} \right)}}{3} + \frac{2 \sqrt{6} \operatorname{atan}{\left(\frac{\sqrt{6}}{3} \right)}}{3} + \log{\left(15 \right)}$$
-log(10) - 2*sqrt(6)*atan(sqrt(6)/2)/3 + 2*sqrt(6)*atan(sqrt(6)/3)/3 + log(15)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.