Sr Examen
Integral de x/(2x^2+x+5) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 | 2*x + x + 5 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(2 x^{2} + x\right) + 5}\, dx$$
Integral(x/(2*x^2 + x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 | 2*x + x + 5 | /
Reescribimos la función subintegral
/ 2*2*x + 1 \
|------------| / -1 \
| 2 | |------|
x \2*x + x + 5/ \4*39/8/
------------ = -------------- + ---------------------------
2 4 2
2*x + x + 5 / ____ ____\
|-4*\/ 39 \/ 39 |
|---------*x - ------| + 1
\ 39 39 / o
/ | | x | ------------ dx | 2 = | 2*x + x + 5 | /
/
|
| 1
2* | --------------------------- dx /
| 2 |
| / ____ ____\ | 2*2*x + 1
| |-4*\/ 39 \/ 39 | | ------------ dx
| |---------*x - ------| + 1 | 2
| \ 39 39 / | 2*x + x + 5
| |
/ /
- ----------------------------------- + ------------------
39 4 En integral
/
|
| 2*2*x + 1
| ------------ dx
| 2
| 2*x + x + 5
|
/
------------------
4 hacemos el cambio
2 u = x + 2*x
entonces
integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
4 4 hacemos cambio inverso
/
|
| 2*2*x + 1
| ------------ dx
| 2
| 2*x + x + 5
| / 2\
/ log\5 + x + 2*x /
------------------ = -----------------
4 4 En integral
/
|
| 1
-2* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x - ------| + 1
| \ 39 39 /
|
/
------------------------------------
39 hacemos el cambio
____ ____
\/ 39 4*x*\/ 39
v = - ------ - ----------
39 39 entonces
integral =
/
|
| 1
-2* | ------ dv
| 2
| 1 + v
|
/ -2*atan(v)
--------------- = ----------
39 39 hacemos cambio inverso
/
|
| 1
-2* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 39 \/ 39 |
| |---------*x - ------| + 1 / ____ ____\
| \ 39 39 / ____ |\/ 39 4*x*\/ 39 |
| -\/ 39 *atan|------ + ----------|
/ \ 39 39 /
------------------------------------ = ----------------------------------
39 78 La solución:
/ ____ ____\
/5 2 x\ ____ |\/ 39 4*x*\/ 39 |
log|- + x + -| \/ 39 *atan|------ + ----------|
\2 2/ \ 39 39 /
C + --------------- - --------------------------------
4 78
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |4*\/ 39 *(1/4 + x)| | / 2\ \/ 39 *atan|------------------| | x log\5 + x + 2*x / \ 39 / | ------------ dx = C + ----------------- - ------------------------------- | 2 4 78 | 2*x + x + 5 | /
$$\int \frac{x}{\left(2 x^{2} + x\right) + 5}\, dx = C + \frac{\log{\left(2 x^{2} + x + 5 \right)}}{4} - \frac{\sqrt{39} \operatorname{atan}{\left(\frac{4 \sqrt{39} \left(x + \frac{1}{4}\right)}{39} \right)}}{78}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |5*\/ 39 | ____ |\/ 39 |
\/ 39 *atan|--------| \/ 39 *atan|------|
log(5/2) log(4) \ 39 / \ 39 /
- -------- + ------ - --------------------- + -------------------
4 4 78 78
$$- \frac{\log{\left(\frac{5}{2} \right)}}{4} - \frac{\sqrt{39} \operatorname{atan}{\left(\frac{5 \sqrt{39}}{39} \right)}}{78} + \frac{\sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{78} + \frac{\log{\left(4 \right)}}{4}$$
=
=
/ ____\ / ____\
____ |5*\/ 39 | ____ |\/ 39 |
\/ 39 *atan|--------| \/ 39 *atan|------|
log(5/2) log(4) \ 39 / \ 39 /
- -------- + ------ - --------------------- + -------------------
4 4 78 78
$$- \frac{\log{\left(\frac{5}{2} \right)}}{4} - \frac{\sqrt{39} \operatorname{atan}{\left(\frac{5 \sqrt{39}}{39} \right)}}{78} + \frac{\sqrt{39} \operatorname{atan}{\left(\frac{\sqrt{39}}{39} \right)}}{78} + \frac{\log{\left(4 \right)}}{4}$$
-log(5/2)/4 + log(4)/4 - sqrt(39)*atan(5*sqrt(39)/39)/78 + sqrt(39)*atan(sqrt(39)/39)/78
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.