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