Sr Examen
Integral de (9*x-13)/(x^2-2*x+6) dx
Solución
Ha introducido
[src]
1 / | | 9*x - 13 | ------------ dx | 2 | x - 2*x + 6 | / 0
$$\int\limits_{0}^{1} \frac{9 x - 13}{\left(x^{2} - 2 x\right) + 6}\, dx$$
Integral((9*x - 13)/(x^2 - 2*x + 6), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 9*x - 13 | ------------ dx | 2 | x - 2*x + 6 | /
Reescribimos la función subintegral
2*x - 2
9*------------ /-4 \
2 |---|
9*x - 13 x - 2*x + 6 \ 5 /
------------ = -------------- + ------------------------
2 2 2
x - 2*x + 6 / ___ ___\
|-\/ 5 \/ 5 |
|-------*x + -----| + 1
\ 5 5 / o
/ | | 9*x - 13 | ------------ dx | 2 = | x - 2*x + 6 | /
/
|
| 1
4* | ------------------------ dx /
| 2 |
| / ___ ___\ | 2*x - 2
| |-\/ 5 \/ 5 | 9* | ------------ dx
| |-------*x + -----| + 1 | 2
| \ 5 5 / | x - 2*x + 6
| |
/ /
- -------------------------------- + --------------------
5 2 En integral
/
|
| 2*x - 2
9* | ------------ dx
| 2
| x - 2*x + 6
|
/
--------------------
2 hacemos el cambio
2 u = x - 2*x
entonces
integral =
/
|
| 1
9* | ----- du
| 6 + u
|
/ 9*log(6 + u)
------------- = ------------
2 2 hacemos cambio inverso
/
|
| 2*x - 2
9* | ------------ dx
| 2
| x - 2*x + 6
| / 2 \
/ 9*log\6 + x - 2*x/
-------------------- = -------------------
2 2 En integral
/
|
| 1
-4* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 5 \/ 5 |
| |-------*x + -----| + 1
| \ 5 5 /
|
/
---------------------------------
5 hacemos el cambio
___ ___
\/ 5 x*\/ 5
v = ----- - -------
5 5 entonces
integral =
/
|
| 1
-4* | ------ dv
| 2
| 1 + v
|
/ -4*atan(v)
--------------- = ----------
5 5 hacemos cambio inverso
/
|
| 1
-4* | ------------------------ dx
| 2
| / ___ ___\
| |-\/ 5 \/ 5 |
| |-------*x + -----| + 1 / ___ ___\
| \ 5 5 / ___ | \/ 5 x*\/ 5 |
| -4*\/ 5 *atan|- ----- + -------|
/ \ 5 5 /
--------------------------------- = --------------------------------
5 5 La solución:
/ ___ ___\
___ | \/ 5 x*\/ 5 |
/ 2 \ 4*\/ 5 *atan|- ----- + -------|
9*log\6 + x - 2*x/ \ 5 5 /
C + ------------------- - -------------------------------
2 5
Respuesta (Indefinida)
[src]
/ ___ \ / ___ |\/ 5 *(-1 + x)| | / 2 \ 4*\/ 5 *atan|--------------| | 9*x - 13 9*log\6 + x - 2*x/ \ 5 / | ------------ dx = C + ------------------- - ---------------------------- | 2 2 5 | x - 2*x + 6 | /
$$\int \frac{9 x - 13}{\left(x^{2} - 2 x\right) + 6}\, dx = C + \frac{9 \log{\left(x^{2} - 2 x + 6 \right)}}{2} - \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5} \left(x - 1\right)}{5} \right)}}{5}$$
Gráfica
Respuesta
[src]
/ ___\
___ |\/ 5 |
4*\/ 5 *atan|-----|
9*log(6) 9*log(5) \ 5 /
- -------- + -------- - -------------------
2 2 5
$$- \frac{9 \log{\left(6 \right)}}{2} - \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{5} + \frac{9 \log{\left(5 \right)}}{2}$$
=
=
/ ___\
___ |\/ 5 |
4*\/ 5 *atan|-----|
9*log(6) 9*log(5) \ 5 /
- -------- + -------- - -------------------
2 2 5
$$- \frac{9 \log{\left(6 \right)}}{2} - \frac{4 \sqrt{5} \operatorname{atan}{\left(\frac{\sqrt{5}}{5} \right)}}{5} + \frac{9 \log{\left(5 \right)}}{2}$$
-9*log(6)/2 + 9*log(5)/2 - 4*sqrt(5)*atan(sqrt(5)/5)/5
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.