Sr Examen
Integral de (x+3)/(x^2+x+2) dx
Solución
Ha introducido
[src]
1 / | | x + 3 | ---------- dx | 2 | x + x + 2 | / 0
$$\int\limits_{0}^{1} \frac{x + 3}{\left(x^{2} + x\right) + 2}\, dx$$
Integral((x + 3)/(x^2 + x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 3 | ---------- dx | 2 | x + x + 2 | /
Reescribimos la función subintegral
/ 2*x + 1 \
|----------| / 5 \
| 2 | |-----|
x + 3 \x + x + 2/ \2*7/4/
---------- = ------------ + -------------------------
2 2 2
x + x + 2 / ___ ___\
|-2*\/ 7 \/ 7 |
|--------*x - -----| + 1
\ 7 7 / o
/ | | x + 3 | ---------- dx | 2 = | x + x + 2 | /
/
|
| 1
/ 10* | ------------------------- dx
| | 2
| 2*x + 1 | / ___ ___\
| ---------- dx | |-2*\/ 7 \/ 7 |
| 2 | |--------*x - -----| + 1
| x + x + 2 | \ 7 7 /
| |
/ /
---------------- + ----------------------------------
2 7 En integral
/
|
| 2*x + 1
| ---------- dx
| 2
| x + x + 2
|
/
----------------
2 hacemos el cambio
2 u = x + x
entonces
integral =
/
|
| 1
| ----- du
| 2 + u
|
/ log(2 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x + 1
| ---------- dx
| 2
| x + x + 2
| / 2\
/ log\2 + x + x /
---------------- = ---------------
2 2 En integral
/
|
| 1
10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1
| \ 7 7 /
|
/
----------------------------------
7 hacemos el cambio
___ ___
\/ 7 2*x*\/ 7
v = - ----- - ---------
7 7 entonces
integral =
/
|
| 1
10* | ------ dv
| 2
| 1 + v
|
/ 10*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
10* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1 / ___ ___\
| \ 7 7 / ___ |\/ 7 2*x*\/ 7 |
| 5*\/ 7 *atan|----- + ---------|
/ \ 7 7 /
---------------------------------- = -------------------------------
7 7 La solución:
/ ___ ___\
___ |\/ 7 2*x*\/ 7 |
/ 2\ 5*\/ 7 *atan|----- + ---------|
log\2 + x + x / \ 7 7 /
C + --------------- + -------------------------------
2 7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(1/2 + x)| | / 2\ 5*\/ 7 *atan|-----------------| | x + 3 log\2 + x + x / \ 7 / | ---------- dx = C + --------------- + ------------------------------- | 2 2 7 | x + x + 2 | /
$$\int \frac{x + 3}{\left(x^{2} + x\right) + 2}\, dx = C + \frac{\log{\left(x^{2} + x + 2 \right)}}{2} + \frac{5 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x + \frac{1}{2}\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
5*\/ 7 *atan|-----| 5*\/ 7 *atan|-------|
log(4) log(2) \ 7 / \ 7 /
------ - ------ - ------------------- + ---------------------
2 2 7 7
$$- \frac{5 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} - \frac{\log{\left(2 \right)}}{2} + \frac{\log{\left(4 \right)}}{2} + \frac{5 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7}$$
=
=
/ ___\ / ___\
___ |\/ 7 | ___ |3*\/ 7 |
5*\/ 7 *atan|-----| 5*\/ 7 *atan|-------|
log(4) log(2) \ 7 / \ 7 /
------ - ------ - ------------------- + ---------------------
2 2 7 7
$$- \frac{5 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} - \frac{\log{\left(2 \right)}}{2} + \frac{\log{\left(4 \right)}}{2} + \frac{5 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7}$$
log(4)/2 - log(2)/2 - 5*sqrt(7)*atan(sqrt(7)/7)/7 + 5*sqrt(7)*atan(3*sqrt(7)/7)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.