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