Sr Examen
Integral de x*dx/(2*x^2+2*x+5) dx
Solución
Ha introducido
[src]
1 / | | x | -------------- dx | 2 | 2*x + 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(2 x^{2} + 2 x\right) + 5}\, dx$$
Integral(x/(2*x^2 + 2*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | -------------- dx | 2 | 2*x + 2*x + 5 | /
Reescribimos la función subintegral
/ 2*2*x + 2 \
|--------------| / -1 \
| 2 | |-----|
x \2*x + 2*x + 5/ \2*9/2/
-------------- = ---------------- + ----------------
2 4 2
2*x + 2*x + 5 / 2*x 1\
|- --- - -| + 1
\ 3 3/ o
/ | | x | -------------- dx | 2 = | 2*x + 2*x + 5 | /
/
|
| 1 /
| ---------------- dx |
| 2 | 2*2*x + 2
| / 2*x 1\ | -------------- dx
| |- --- - -| + 1 | 2
| \ 3 3/ | 2*x + 2*x + 5
| |
/ /
- ---------------------- + --------------------
9 4 En integral
/
|
| 2*2*x + 2
| -------------- dx
| 2
| 2*x + 2*x + 5
|
/
--------------------
4 hacemos el cambio
2 u = 2*x + 2*x
entonces
integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
4 4 hacemos cambio inverso
/
|
| 2*2*x + 2
| -------------- dx
| 2
| 2*x + 2*x + 5
| / 2\
/ log\5 + 2*x + 2*x /
-------------------- = -------------------
4 4 En integral
/
|
| 1
- | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/
|
/
------------------------
9 hacemos el cambio
1 2*x
v = - - - ---
3 3 entonces
integral =
/
|
| 1
- | ------ dv
| 2
| 1 + v
|
/ -atan(v)
-------------- = ---------
9 9 hacemos cambio inverso
/
|
| 1
- | ---------------- dx
| 2
| / 2*x 1\
| |- --- - -| + 1
| \ 3 3/ /1 2*x\
| -atan|- + ---|
/ \3 3 /
------------------------ = ---------------
9 6 La solución:
/1 2*x\ /5 2\
atan|- + ---| log|- + x + x |
\3 3 / \2 /
C - ------------- + ---------------
6 4
Respuesta (Indefinida)
[src]
/ /1 2*x\ | atan|- + ---| / 2\ | x \3 3 / log\5 + 2*x + 2*x / | -------------- dx = C - ------------- + ------------------- | 2 6 4 | 2*x + 2*x + 5 | /
$$\int \frac{x}{\left(2 x^{2} + 2 x\right) + 5}\, dx = C + \frac{\log{\left(2 x^{2} + 2 x + 5 \right)}}{4} - \frac{\operatorname{atan}{\left(\frac{2 x}{3} + \frac{1}{3} \right)}}{6}$$
Gráfica
Respuesta
[src]
log(5/2) pi log(9/2) atan(1/3)
- -------- - -- + -------- + ---------
4 24 4 6
$$- \frac{\log{\left(\frac{5}{2} \right)}}{4} - \frac{\pi}{24} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{6} + \frac{\log{\left(\frac{9}{2} \right)}}{4}$$
=
=
log(5/2) pi log(9/2) atan(1/3)
- -------- - -- + -------- + ---------
4 24 4 6
$$- \frac{\log{\left(\frac{5}{2} \right)}}{4} - \frac{\pi}{24} + \frac{\operatorname{atan}{\left(\frac{1}{3} \right)}}{6} + \frac{\log{\left(\frac{9}{2} \right)}}{4}$$
-log(5/2)/4 - pi/24 + log(9/2)/4 + atan(1/3)/6
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.