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