Sr Examen
Integral de x/(2*x^2+0.5*x+2.5) dx
Solución
Ha introducido
[src]
1 / | | x | ------------ dx | 2 x 5 | 2*x + - + - | 2 2 | / 0
$$\int\limits_{0}^{1} \frac{x}{\left(2 x^{2} + \frac{x}{2}\right) + \frac{5}{2}}\, dx$$
Integral(x/(2*x^2 + x/2 + 5/2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | x | ------------ dx | 2 x 5 | 2*x + - + - | 2 2 | /
Reescribimos la función subintegral
/2*2*x + 1/2 \ /-1 \
|------------| |----|
| 2 x 5| | 79|
|2*x + - + -| |8*--|
x \ 2 2/ \ 32/
------------ = -------------- + ---------------------------
2 x 5 4 2
2*x + - + - / ____ ____\
2 2 |-8*\/ 79 \/ 79 |
|---------*x - ------| + 1
\ 79 79 / o
/ | | x | ------------ dx | 2 x 5 = | 2*x + - + - | 2 2 | /
/
|
| 1 /
4* | --------------------------- dx |
| 2 | 2*2*x + 1/2
| / ____ ____\ | ------------ dx
| |-8*\/ 79 \/ 79 | | 2 x 5
| |---------*x - ------| + 1 | 2*x + - + -
| \ 79 79 / | 2 2
| |
/ /
- ----------------------------------- + ------------------
79 4 En integral
/
|
| 2*2*x + 1/2
| ------------ dx
| 2 x 5
| 2*x + - + -
| 2 2
|
/
------------------
4 hacemos el cambio
x 2
u = - + 2*x
2 entonces
integral =
/
|
| 1
| ------- du
| 5/2 + u
|
/ log(5 + 2*u)
------------- = ------------
4 4 hacemos cambio inverso
/
|
| 2*2*x + 1/2
| ------------ dx
| 2 x 5
| 2*x + - + -
| 2 2
| / 2\
/ log\5 + x + 4*x /
------------------ = -----------------
4 4 En integral
/
|
| 1
-4* | --------------------------- dx
| 2
| / ____ ____\
| |-8*\/ 79 \/ 79 |
| |---------*x - ------| + 1
| \ 79 79 /
|
/
------------------------------------
79 hacemos el cambio
____ ____
\/ 79 8*x*\/ 79
v = - ------ - ----------
79 79 entonces
integral =
/
|
| 1
-4* | ------ dv
| 2
| 1 + v
|
/ -4*atan(v)
--------------- = ----------
79 79 hacemos cambio inverso
/
|
| 1
-4* | --------------------------- dx
| 2
| / ____ ____\
| |-8*\/ 79 \/ 79 |
| |---------*x - ------| + 1 / ____ ____\
| \ 79 79 / ____ |\/ 79 8*x*\/ 79 |
| -\/ 79 *atan|------ + ----------|
/ \ 79 79 /
------------------------------------ = ----------------------------------
79 158 La solución:
/ ____ ____\
/5 2 x\ ____ |\/ 79 8*x*\/ 79 |
log|- + x + -| \/ 79 *atan|------ + ----------|
\4 4/ \ 79 79 /
C + --------------- - --------------------------------
4 158
Respuesta (Indefinida)
[src]
/ ____ \ / /5 x 2\ ____ |8*\/ 79 *(1/8 + x)| | log|- + - + 2*x | \/ 79 *atan|------------------| | x \2 2 / \ 79 / | ------------ dx = C + ----------------- - ------------------------------- | 2 x 5 4 158 | 2*x + - + - | 2 2 | /
$$\int \frac{x}{\left(2 x^{2} + \frac{x}{2}\right) + \frac{5}{2}}\, dx = C + \frac{\log{\left(2 x^{2} + \frac{x}{2} + \frac{5}{2} \right)}}{4} - \frac{\sqrt{79} \operatorname{atan}{\left(\frac{8 \sqrt{79} \left(x + \frac{1}{8}\right)}{79} \right)}}{158}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |9*\/ 79 | ____ |\/ 79 |
\/ 79 *atan|--------| \/ 79 *atan|------|
log(5/4) log(5/2) \ 79 / \ 79 /
- -------- + -------- - --------------------- + -------------------
4 4 158 158
$$- \frac{\log{\left(\frac{5}{4} \right)}}{4} - \frac{\sqrt{79} \operatorname{atan}{\left(\frac{9 \sqrt{79}}{79} \right)}}{158} + \frac{\sqrt{79} \operatorname{atan}{\left(\frac{\sqrt{79}}{79} \right)}}{158} + \frac{\log{\left(\frac{5}{2} \right)}}{4}$$
=
=
/ ____\ / ____\
____ |9*\/ 79 | ____ |\/ 79 |
\/ 79 *atan|--------| \/ 79 *atan|------|
log(5/4) log(5/2) \ 79 / \ 79 /
- -------- + -------- - --------------------- + -------------------
4 4 158 158
$$- \frac{\log{\left(\frac{5}{4} \right)}}{4} - \frac{\sqrt{79} \operatorname{atan}{\left(\frac{9 \sqrt{79}}{79} \right)}}{158} + \frac{\sqrt{79} \operatorname{atan}{\left(\frac{\sqrt{79}}{79} \right)}}{158} + \frac{\log{\left(\frac{5}{2} \right)}}{4}$$
-log(5/4)/4 + log(5/2)/4 - sqrt(79)*atan(9*sqrt(79)/79)/158 + sqrt(79)*atan(sqrt(79)/79)/158
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.