Sr Examen
Integral de x/(x^2+3x+4) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | x + 3*x + 4 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(x^{2} + 3 x\right) + 4}\, dx$$
Integral(x/(x^2 + 3*x + 4), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 | x + 3*x + 4 | /
Reescribimos la función subintegral
/ 2*x + 3 \
|------------| / -3 \
| 2 | |-----|
x \x + 3*x + 4/ \2*7/4/
------------ = -------------- + ---------------------------
2 2 2
x + 3*x + 4 / ___ ___\
|-2*\/ 7 3*\/ 7 |
|--------*x - -------| + 1
\ 7 7 / o
/ | | x | ------------ dx | 2 = | x + 3*x + 4 | /
/
|
| 1
/ 6* | --------------------------- dx
| | 2
| 2*x + 3 | / ___ ___\
| ------------ dx | |-2*\/ 7 3*\/ 7 |
| 2 | |--------*x - -------| + 1
| x + 3*x + 4 | \ 7 7 /
| |
/ /
------------------ - -----------------------------------
2 7 En integral
/
|
| 2*x + 3
| ------------ dx
| 2
| x + 3*x + 4
|
/
------------------
2 hacemos el cambio
2 u = x + 3*x
entonces
integral =
/
|
| 1
| ----- du
| 4 + u
|
/ log(4 + u)
----------- = ----------
2 2 hacemos cambio inverso
/
|
| 2*x + 3
| ------------ dx
| 2
| x + 3*x + 4
| / 2 \
/ log\4 + x + 3*x/
------------------ = -----------------
2 2 En integral
/
|
| 1
-6* | --------------------------- dx
| 2
| / ___ ___\
| |-2*\/ 7 3*\/ 7 |
| |--------*x - -------| + 1
| \ 7 7 /
|
/
------------------------------------
7 hacemos el cambio
___ ___
3*\/ 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 3*\/ 7 |
| |--------*x - -------| + 1 / ___ ___\
| \ 7 7 / ___ |3*\/ 7 2*x*\/ 7 |
| -3*\/ 7 *atan|------- + ---------|
/ \ 7 7 /
------------------------------------ = ----------------------------------
7 7 La solución:
/ ___ ___\
___ |3*\/ 7 2*x*\/ 7 |
/ 2 \ 3*\/ 7 *atan|------- + ---------|
log\4 + x + 3*x/ \ 7 7 /
C + ----------------- - ---------------------------------
2 7
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |2*\/ 7 *(3/2 + x)| | / 2 \ 3*\/ 7 *atan|-----------------| | x log\4 + x + 3*x/ \ 7 / | ------------ dx = C + ----------------- - ------------------------------- | 2 2 7 | x + 3*x + 4 | /
$$\int \frac{x}{\left(x^{2} + 3 x\right) + 4}\, dx = C + \frac{\log{\left(x^{2} + 3 x + 4 \right)}}{2} - \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{2 \sqrt{7} \left(x + \frac{3}{2}\right)}{7} \right)}}{7}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
___ |5*\/ 7 | ___ |3*\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-------|
log(8) log(4) \ 7 / \ 7 /
------ - ------ - --------------------- + ---------------------
2 2 7 7
$$- \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7} - \frac{\log{\left(4 \right)}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{\log{\left(8 \right)}}{2}$$
=
=
/ ___\ / ___\
___ |5*\/ 7 | ___ |3*\/ 7 |
3*\/ 7 *atan|-------| 3*\/ 7 *atan|-------|
log(8) log(4) \ 7 / \ 7 /
------ - ------ - --------------------- + ---------------------
2 2 7 7
$$- \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{5 \sqrt{7}}{7} \right)}}{7} - \frac{\log{\left(4 \right)}}{2} + \frac{3 \sqrt{7} \operatorname{atan}{\left(\frac{3 \sqrt{7}}{7} \right)}}{7} + \frac{\log{\left(8 \right)}}{2}$$
log(8)/2 - log(4)/2 - 3*sqrt(7)*atan(5*sqrt(7)/7)/7 + 3*sqrt(7)*atan(3*sqrt(7)/7)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.