Sr Examen
Integral de (3x+2)/(x^2+x+4) dx
Solución
Ha introducido
[src]
1 / | | 3*x + 2 | ---------- dx | 2 | x + x + 4 | / 0
$$\int\limits_{0}^{1} \frac{3 x + 2}{\left(x^{2} + x\right) + 4}\, dx$$
Integral((3*x + 2)/(x^2 + x + 4), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 3*x + 2 | ---------- dx | 2 | x + x + 4 | /
Reescribimos la función subintegral
2*x + 1
3*---------- / 1 \
2 |------|
3*x + 2 x + x + 4 \2*15/4/
---------- = ------------ + ---------------------------
2 2 2
x + x + 4 / ____ ____\
|-2*\/ 15 \/ 15 |
|---------*x - ------| + 1
\ 15 15 / o
/ | | 3*x + 2 | ---------- dx | 2 = | x + x + 4 | /
/
|
| 1
2* | --------------------------- dx /
| 2 |
| / ____ ____\ | 2*x + 1
| |-2*\/ 15 \/ 15 | 3* | ---------- dx
| |---------*x - ------| + 1 | 2
| \ 15 15 / | x + x + 4
| |
/ /
----------------------------------- + ------------------
15 2 En integral
/
|
| 2*x + 1
3* | ---------- dx
| 2
| x + x + 4
|
/
------------------
2 hacemos el cambio
2 u = x + x
entonces
integral =
/
|
| 1
3* | ----- du
| 4 + u
|
/ 3*log(4 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x + 1
3* | ---------- dx
| 2
| x + x + 4
| / 2\
/ 3*log\4 + x + x /
------------------ = -----------------
2 2 En integral
/
|
| 1
2* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x - ------| + 1
| \ 15 15 /
|
/
-----------------------------------
15 hacemos el cambio
____ ____
\/ 15 2*x*\/ 15
v = - ------ - ----------
15 15 entonces
integral =
/
|
| 1
2* | ------ dv
| 2
| 1 + v
|
/ 2*atan(v)
-------------- = ---------
15 15 hacemos cambio inverso
/
|
| 1
2* | --------------------------- dx
| 2
| / ____ ____\
| |-2*\/ 15 \/ 15 |
| |---------*x - ------| + 1 / ____ ____\
| \ 15 15 / ____ |\/ 15 2*x*\/ 15 |
| \/ 15 *atan|------ + ----------|
/ \ 15 15 /
----------------------------------- = --------------------------------
15 15 La solución:
/ ____ ____\
____ |\/ 15 2*x*\/ 15 |
/ 2\ \/ 15 *atan|------ + ----------|
3*log\4 + x + x / \ 15 15 /
C + ----------------- + --------------------------------
2 15
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 15 *(1/2 + x)| | / 2\ \/ 15 *atan|------------------| | 3*x + 2 3*log\4 + x + x / \ 15 / | ---------- dx = C + ----------------- + ------------------------------- | 2 2 15 | x + x + 4 | /
$$\int \frac{3 x + 2}{\left(x^{2} + x\right) + 4}\, dx = C + \frac{3 \log{\left(x^{2} + x + 4 \right)}}{2} + \frac{\sqrt{15} \operatorname{atan}{\left(\frac{2 \sqrt{15} \left(x + \frac{1}{2}\right)}{15} \right)}}{15}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
\/ 15 *atan|------| \/ 15 *atan|------|
3*log(4) 3*log(6) \ 15 / \ 5 /
- -------- + -------- - ------------------- + -------------------
2 2 15 15
$$- \frac{3 \log{\left(4 \right)}}{2} - \frac{\sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{\sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{15} + \frac{3 \log{\left(6 \right)}}{2}$$
=
=
/ ____\ / ____\
____ |\/ 15 | ____ |\/ 15 |
\/ 15 *atan|------| \/ 15 *atan|------|
3*log(4) 3*log(6) \ 15 / \ 5 /
- -------- + -------- - ------------------- + -------------------
2 2 15 15
$$- \frac{3 \log{\left(4 \right)}}{2} - \frac{\sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)}}{15} + \frac{\sqrt{15} \operatorname{atan}{\left(\frac{\sqrt{15}}{5} \right)}}{15} + \frac{3 \log{\left(6 \right)}}{2}$$
-3*log(4)/2 + 3*log(6)/2 - sqrt(15)*atan(sqrt(15)/15)/15 + sqrt(15)*atan(sqrt(15)/5)/15
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.