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