Sr Examen
Integral de (5x+1)/(x^2+x+2) dx
Solución
Ha introducido
[src]
1 / | | 5*x + 1 | ---------- dx | 2 | x + x + 2 | / 0
$$\int\limits_{0}^{1} \frac{5 x + 1}{\left(x^{2} + x\right) + 2}\, dx$$
Integral((5*x + 1)/(x^2 + x + 2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 5*x + 1 | ---------- dx | 2 | x + x + 2 | /
Reescribimos la función subintegral
2*x + 1
5*---------- / -3 \
2 |-----|
5*x + 1 x + x + 2 \2*7/4/
---------- = ------------ + -------------------------
2 2 2
x + x + 2 / ___ ___\
|-2*\/ 7 \/ 7 |
|--------*x - -----| + 1
\ 7 7 / o
/ | | 5*x + 1 | ---------- dx | 2 = | x + x + 2 | /
/
|
| 1
6* | ------------------------- dx /
| 2 |
| / ___ ___\ | 2*x + 1
| |-2*\/ 7 \/ 7 | 5* | ---------- dx
| |--------*x - -----| + 1 | 2
| \ 7 7 / | x + x + 2
| |
/ /
- --------------------------------- + ------------------
7 2 En integral
/
|
| 2*x + 1
5* | ---------- dx
| 2
| x + x + 2
|
/
------------------
2 hacemos el cambio
2 u = x + x
entonces
integral =
/
|
| 1
5* | ----- du
| 2 + u
|
/ 5*log(2 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 1
5* | ---------- dx
| 2
| x + x + 2
| / 2\
/ 5*log\2 + x + x /
------------------ = -----------------
2 2 En integral
/
|
| 1
-6* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1
| \ 7 7 /
|
/
----------------------------------
7 hacemos el cambio
___ ___
\/ 7 2*x*\/ 7
v = - ----- - ---------
7 7 entonces
integral =
/
|
| 1
-6* | ------ dv
| 2
| 1 + v
|
/ -6*atan(v)
--------------- = ----------
7 7 hacemos cambio inverso
/
|
| 1
-6* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 \/ 7 |
| |--------*x - -----| + 1 / ___ ___\
| \ 7 7 / ___ |\/ 7 2*x*\/ 7 |
| -3*\/ 7 *atan|----- + ---------|
/ \ 7 7 /
---------------------------------- = --------------------------------
7 7 La solución:
/ ___ ___\
___ |\/ 7 2*x*\/ 7 |
/ 2\ 3*\/ 7 *atan|----- + ---------|
5*log\2 + x + x / \ 7 7 /
C + ----------------- - -------------------------------
2 7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(1/2 + x)| | / 2\ 3*\/ 7 *atan|-----------------| | 5*x + 1 5*log\2 + x + x / \ 7 / | ---------- dx = C + ----------------- - ------------------------------- | 2 2 7 | x + x + 2 | /
$$\int \frac{5 x + 1}{\left(x^{2} + x\right) + 2}\, dx = C + \frac{5 \log{\left(x^{2} + x + 2 \right)}}{2} - \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x + \frac{1}{2}\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |3*\/ 7 | ___ |\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-----|
5*log(2) 5*log(4) \ 7 / \ 7 /
- -------- + -------- - --------------------- + -------------------
2 2 7 7
$$- \frac{5 \log{\left(2 \right)}}{2} - \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \frac{5 \log{\left(4 \right)}}{2}$$
=
=
/ ___\ / ___\
___ |3*\/ 7 | ___ |\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-----|
5*log(2) 5*log(4) \ 7 / \ 7 /
- -------- + -------- - --------------------- + -------------------
2 2 7 7
$$- \frac{5 \log{\left(2 \right)}}{2} - \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{\sqrt{7}}{7} \right)}}{7} + \frac{5 \log{\left(4 \right)}}{2}$$
-5*log(2)/2 + 5*log(4)/2 - 3*sqrt(7)*atan(3*sqrt(7)/7)/7 + 3*sqrt(7)*atan(sqrt(7)/7)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.