Sr Examen
Integral de (x+5)/(2x^2+2x+3) dx
Solución
Ha introducido
[src]
1 / | | x + 5 | -------------- dx | 2 | 2*x + 2*x + 3 | / 0
$$\int\limits_{0}^{1} \frac{x + 5}{\left(2 x^{2} + 2 x\right) + 3}\, dx$$
Integral((x + 5)/(2*x^2 + 2*x + 3), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x + 5 | -------------- dx | 2 | 2*x + 2*x + 3 | /
Reescribimos la función subintegral
/ 2*2*x + 2 \
|--------------| / 9 \
| 2 | |-----|
x + 5 \2*x + 2*x + 3/ \2*5/2/
-------------- = ---------------- + -------------------------
2 4 2
2*x + 2*x + 3 / ___ ___\
|-2*\/ 5 \/ 5 |
|--------*x - -----| + 1
\ 5 5 / o
/ | | x + 5 | -------------- dx | 2 = | 2*x + 2*x + 3 | /
/
|
| 1
/ 9* | ------------------------- dx
| | 2
| 2*2*x + 2 | / ___ ___\
| -------------- dx | |-2*\/ 5 \/ 5 |
| 2 | |--------*x - -----| + 1
| 2*x + 2*x + 3 | \ 5 5 /
| |
/ /
-------------------- + ---------------------------------
4 5 En integral
/
|
| 2*2*x + 2
| -------------- dx
| 2
| 2*x + 2*x + 3
|
/
--------------------
4 hacemos el cambio
2 u = 2*x + 2*x
entonces
integral =
/
|
| 1
| ----- du
| 3 + u
|
/ log(3 + u)
----------- = ----------
4 4 hacemos cambio inverso
/
|
| 2*2*x + 2
| -------------- dx
| 2
| 2*x + 2*x + 3
| / 2\
/ log\3 + 2*x + 2*x /
-------------------- = -------------------
4 4 En integral
/
|
| 1
9* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 5 \/ 5 |
| |--------*x - -----| + 1
| \ 5 5 /
|
/
---------------------------------
5 hacemos el cambio
___ ___
\/ 5 2*x*\/ 5
v = - ----- - ---------
5 5 entonces
integral =
/
|
| 1
9* | ------ dv
| 2
| 1 + v
|
/ 9*atan(v)
-------------- = ---------
5 5 hacemos cambio inverso
/
|
| 1
9* | ------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 5 \/ 5 |
| |--------*x - -----| + 1 / ___ ___\
| \ 5 5 / ___ |\/ 5 2*x*\/ 5 |
| 9*\/ 5 *atan|----- + ---------|
/ \ 5 5 /
--------------------------------- = -------------------------------
5 10 La solución:
/ ___ ___\
/3 2\ ___ |\/ 5 2*x*\/ 5 |
log|- + x + x | 9*\/ 5 *atan|----- + ---------|
\2 / \ 5 5 /
C + --------------- + -------------------------------
4 10
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 5 *(1/2 + x)| | / 2\ 9*\/ 5 *atan|-----------------| | x + 5 log\3 + 2*x + 2*x / \ 5 / | -------------- dx = C + ------------------- + ------------------------------- | 2 4 10 | 2*x + 2*x + 3 | /
$$\int \frac{x + 5}{\left(2 x^{2} + 2 x\right) + 3}\, dx = C + \frac{\log{\left(2 x^{2} + 2 x + 3 \right)}}{4} + \frac{9 \sqrt{5} \operatorname{atan}{\left(\frac{2 \sqrt{5} \left(x + \frac{1}{2}\right)}{5} \right)}}{10}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |\/ 5 | ___ |3*\/ 5 |
9*\/ 5 *atan|-----| 9*\/ 5 *atan|-------|
log(3/2) log(7/2) \ 5 / \ 5 /
- -------- + -------- - ------------------- + ---------------------
4 4 10 10
$$- \frac{9 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{10} - \frac{\log{\left(\frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{7}{2} \right)}}{4} + \frac{9 \sqrt{5} \operatorname{atan}{\left(\frac{3 \sqrt{5}}{5} \right)}}{10}$$
=
=
/ ___\ / ___\
___ |\/ 5 | ___ |3*\/ 5 |
9*\/ 5 *atan|-----| 9*\/ 5 *atan|-------|
log(3/2) log(7/2) \ 5 / \ 5 /
- -------- + -------- - ------------------- + ---------------------
4 4 10 10
$$- \frac{9 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{10} - \frac{\log{\left(\frac{3}{2} \right)}}{4} + \frac{\log{\left(\frac{7}{2} \right)}}{4} + \frac{9 \sqrt{5} \operatorname{atan}{\left(\frac{3 \sqrt{5}}{5} \right)}}{10}$$
-log(3/2)/4 + log(7/2)/4 - 9*sqrt(5)*atan(sqrt(5)/5)/10 + 9*sqrt(5)*atan(3*sqrt(5)/5)/10
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.