Sr Examen
Integral de (2*x)/(4*(x*x)+4*x+17) dx
Solución
Ha introducido
[src]
0 / | | 2*x | ---------------- dx | 4*x*x + 4*x + 17 | / -1/2
$$\int\limits_{- \frac{1}{2}}^{0} \frac{2 x}{\left(4 x x + 4 x\right) + 17}\, dx$$
Integral((2*x)/(4*(x*x) + 4*x + 17), (x, -1/2, 0))
Solución detallada
Tenemos el integral:
/ | | 2*x | ---------------- dx | 4*x*x + 4*x + 17 | /
Reescribimos la función subintegral
/ 4*2*x + 4 \
|---------------| /-1 \
| 2 | |---|
2*x \4*x + 4*x + 17/ \ 16/
---------------- = ----------------- + --------------
4*x*x + 4*x + 17 4 2
/ x 1\
|- - - -| + 1
\ 2 4/ o
/ | | 2*x | ---------------- dx = | 4*x*x + 4*x + 17 | /
/
|
| 1 /
| -------------- dx |
| 2 | 4*2*x + 4
| / x 1\ | --------------- dx
| |- - - -| + 1 | 2
| \ 2 4/ | 4*x + 4*x + 17
| |
/ /
- -------------------- + ---------------------
16 4 En integral
/
|
| 4*2*x + 4
| --------------- dx
| 2
| 4*x + 4*x + 17
|
/
---------------------
4 hacemos el cambio
2 u = 4*x + 4*x
entonces
integral =
/
|
| 1
| ------ du
| 17 + u
|
/ log(17 + u)
------------ = -----------
4 4 hacemos cambio inverso
/
|
| 4*2*x + 4
| --------------- dx
| 2
| 4*x + 4*x + 17
| / 2\
/ log\17 + 4*x + 4*x /
--------------------- = --------------------
4 4 En integral
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 4/
|
/
----------------------
16 hacemos el cambio
1 x
v = - - - -
4 2entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
16 16 hacemos cambio inverso
/
|
| 1
- | -------------- dx
| 2
| / x 1\
| |- - - -| + 1
| \ 2 4/ /1 x\
| -atan|- + -|
/ \4 2/
---------------------- = -------------
16 8 La solución:
/1 x\ /17 2\
atan|- + -| log|-- + x + x |
\4 2/ \4 /
C - ----------- + ----------------
8 4
Respuesta (Indefinida)
[src]
/ /1 x\ | atan|- + -| / 2\ | 2*x \4 2/ log\17 + 4*x + 4*x / | ---------------- dx = C - ----------- + -------------------- | 4*x*x + 4*x + 17 8 4 | /
$$\int \frac{2 x}{\left(4 x x + 4 x\right) + 17}\, dx = C + \frac{\log{\left(4 x^{2} + 4 x + 17 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{4} \right)}}{8}$$
Gráfica
Respuesta
[src]
log(4) atan(1/4) log(17/4)
- ------ - --------- + ---------
4 8 4
$$- \frac{\log{\left(4 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{1}{4} \right)}}{8} + \frac{\log{\left(\frac{17}{4} \right)}}{4}$$
=
=
log(4) atan(1/4) log(17/4)
- ------ - --------- + ---------
4 8 4
$$- \frac{\log{\left(4 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{1}{4} \right)}}{8} + \frac{\log{\left(\frac{17}{4} \right)}}{4}$$
-log(4)/4 - atan(1/4)/8 + log(17/4)/4
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.