Sr Examen
Integral de (2x-1)/(4x^2+2x+5) dx
Solución
Ha introducido
[src]
1 / | | 2*x - 1 | -------------- dx | 2 | 4*x + 2*x + 5 | / 0
$$\int\limits_{0}^{1} \frac{2 x - 1}{\left(4 x^{2} + 2 x\right) + 5}\, dx$$
Integral((2*x - 1)/(4*x^2 + 2*x + 5), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 2*x - 1 | -------------- dx | 2 | 4*x + 2*x + 5 | /
Reescribimos la función subintegral
/ 4*2*x + 2 \
|--------------| / -3 \
| 2 | |------|
2*x - 1 \4*x + 2*x + 5/ \2*19/4/
-------------- = ---------------- + ---------------------------
2 4 2
4*x + 2*x + 5 / ____ ____\
|-4*\/ 19 \/ 19 |
|---------*x - ------| + 1
\ 19 19 / o
/ | | 2*x - 1 | -------------- dx | 2 = | 4*x + 2*x + 5 | /
/
|
| 1
6* | --------------------------- dx /
| 2 |
| / ____ ____\ | 4*2*x + 2
| |-4*\/ 19 \/ 19 | | -------------- dx
| |---------*x - ------| + 1 | 2
| \ 19 19 / | 4*x + 2*x + 5
| |
/ /
- ----------------------------------- + --------------------
19 4 En integral
/
|
| 4*2*x + 2
| -------------- dx
| 2
| 4*x + 2*x + 5
|
/
--------------------
4 hacemos el cambio
2 u = 2*x + 4*x
entonces
integral =
/
|
| 1
| ----- du
| 5 + u
|
/ log(5 + u)
----------- = ----------
4 4 hacemos cambio inverso
/
|
| 4*2*x + 2
| -------------- dx
| 2
| 4*x + 2*x + 5
| / 2\
/ log\5 + 2*x + 4*x /
-------------------- = -------------------
4 4 En integral
/
|
| 1
-6* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 19 \/ 19 |
| |---------*x - ------| + 1
| \ 19 19 /
|
/
------------------------------------
19 hacemos el cambio
____ ____
\/ 19 4*x*\/ 19
v = - ------ - ----------
19 19 entonces
integral =
/
|
| 1
-6* | ------ dv
| 2
| 1 + v
|
/ -6*atan(v)
--------------- = ----------
19 19 hacemos cambio inverso
/
|
| 1
-6* | --------------------------- dx
| 2
| / ____ ____\
| |-4*\/ 19 \/ 19 |
| |---------*x - ------| + 1 / ____ ____\
| \ 19 19 / ____ |\/ 19 4*x*\/ 19 |
| -3*\/ 19 *atan|------ + ----------|
/ \ 19 19 /
------------------------------------ = -----------------------------------
19 38 La solución:
/ ____ ____\
/5 2 x\ ____ |\/ 19 4*x*\/ 19 |
log|- + x + -| 3*\/ 19 *atan|------ + ----------|
\4 2/ \ 19 19 /
C + --------------- - ----------------------------------
4 38
Respuesta (Indefinida)
[src]
/ ____ \ / ____ |2*\/ 19 *(1/2 + 2*x)| | / 2\ 3*\/ 19 *atan|--------------------| | 2*x - 1 log\10 + 4*x + 8*x / \ 19 / | -------------- dx = C + -------------------- - ----------------------------------- | 2 4 38 | 4*x + 2*x + 5 | /
$$\int \frac{2 x - 1}{\left(4 x^{2} + 2 x\right) + 5}\, dx = C + \frac{\log{\left(8 x^{2} + 4 x + 10 \right)}}{4} - \frac{3 \sqrt{19} \operatorname{atan}{\left(\frac{2 \sqrt{19} \left(2 x + \frac{1}{2}\right)}{19} \right)}}{38}$$
Gráfica
Respuesta
[src]
/ ____\ / ____\
____ |5*\/ 19 | ____ |\/ 19 |
3*\/ 19 *atan|--------| 3*\/ 19 *atan|------|
log(5/4) log(11/4) \ 19 / \ 19 /
- -------- + --------- - ----------------------- + ---------------------
4 4 38 38
$$- \frac{3 \sqrt{19} \operatorname{atan}{\left(\frac{5 \sqrt{19}}{19} \right)}}{38} - \frac{\log{\left(\frac{5}{4} \right)}}{4} + \frac{3 \sqrt{19} \operatorname{atan}{\left(\frac{\sqrt{19}}{19} \right)}}{38} + \frac{\log{\left(\frac{11}{4} \right)}}{4}$$
=
=
/ ____\ / ____\
____ |5*\/ 19 | ____ |\/ 19 |
3*\/ 19 *atan|--------| 3*\/ 19 *atan|------|
log(5/4) log(11/4) \ 19 / \ 19 /
- -------- + --------- - ----------------------- + ---------------------
4 4 38 38
$$- \frac{3 \sqrt{19} \operatorname{atan}{\left(\frac{5 \sqrt{19}}{19} \right)}}{38} - \frac{\log{\left(\frac{5}{4} \right)}}{4} + \frac{3 \sqrt{19} \operatorname{atan}{\left(\frac{\sqrt{19}}{19} \right)}}{38} + \frac{\log{\left(\frac{11}{4} \right)}}{4}$$
-log(5/4)/4 + log(11/4)/4 - 3*sqrt(19)*atan(5*sqrt(19)/19)/38 + 3*sqrt(19)*atan(sqrt(19)/19)/38
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.