Sr Examen
Integral de (5x+3)/(x^2-10x+34) dx
Solución
Ha introducido
[src]
1 / | | 5*x + 3 | -------------- dx | 2 | x - 10*x + 34 | / 0
$$\int\limits_{0}^{1} \frac{5 x + 3}{\left(x^{2} - 10 x\right) + 34}\, dx$$
Integral((5*x + 3)/(x^2 - 10*x + 34), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | 5*x + 3 | -------------- dx | 2 | x - 10*x + 34 | /
Reescribimos la función subintegral
2*x - 10
5*-------------- /28\
2 |--|
5*x + 3 x - 10*x + 34 \9 /
-------------- = ---------------- + --------------
2 2 2
x - 10*x + 34 / x 5\
|- - + -| + 1
\ 3 3/ o
/ | | 5*x + 3 | -------------- dx | 2 = | x - 10*x + 34 | /
/
|
/ | 1
| 28* | -------------- dx
| 2*x - 10 | 2
5* | -------------- dx | / x 5\
| 2 | |- - + -| + 1
| x - 10*x + 34 | \ 3 3/
| |
/ /
---------------------- + -----------------------
2 9 En integral
/
|
| 2*x - 10
5* | -------------- dx
| 2
| x - 10*x + 34
|
/
----------------------
2 hacemos el cambio
2 u = x - 10*x
entonces
integral =
/
|
| 1
5* | ------ du
| 34 + u
|
/ 5*log(34 + u)
-------------- = -------------
2 2 hacemos cambio inverso
/
|
| 2*x - 10
5* | -------------- dx
| 2
| x - 10*x + 34
| / 2 \
/ 5*log\34 + x - 10*x/
---------------------- = ---------------------
2 2 En integral
/
|
| 1
28* | -------------- dx
| 2
| / x 5\
| |- - + -| + 1
| \ 3 3/
|
/
-----------------------
9 hacemos el cambio
5 x
v = - - -
3 3entonces
integral =
/
|
| 1
28* | ------ dv
| 2
| 1 + v
|
/ 28*atan(v)
--------------- = ----------
9 9 hacemos cambio inverso
/
|
| 1
28* | -------------- dx
| 2
| / x 5\
| |- - + -| + 1
| \ 3 3/ / 5 x\
| 28*atan|- - + -|
/ \ 3 3/
----------------------- = ----------------
9 3 La solución:
/ 5 x\
/ 2 \ 28*atan|- - + -|
5*log\34 + x - 10*x/ \ 3 3/
C + --------------------- + ----------------
2 3
Respuesta (Indefinida)
[src]
/ / 5 x\ | / 2 \ 28*atan|- - + -| | 5*x + 3 5*log\34 + x - 10*x/ \ 3 3/ | -------------- dx = C + --------------------- + ---------------- | 2 2 3 | x - 10*x + 34 | /
$$\int \frac{5 x + 3}{\left(x^{2} - 10 x\right) + 34}\, dx = C + \frac{5 \log{\left(x^{2} - 10 x + 34 \right)}}{2} + \frac{28 \operatorname{atan}{\left(\frac{x}{3} - \frac{5}{3} \right)}}{3}$$
Gráfica
Respuesta
[src]
28*atan(4/3) 5*log(34) 5*log(25) 28*atan(5/3)
- ------------ - --------- + --------- + ------------
3 2 2 3
$$- \frac{5 \log{\left(34 \right)}}{2} - \frac{28 \operatorname{atan}{\left(\frac{4}{3} \right)}}{3} + \frac{5 \log{\left(25 \right)}}{2} + \frac{28 \operatorname{atan}{\left(\frac{5}{3} \right)}}{3}$$
=
=
28*atan(4/3) 5*log(34) 5*log(25) 28*atan(5/3)
- ------------ - --------- + --------- + ------------
3 2 2 3
$$- \frac{5 \log{\left(34 \right)}}{2} - \frac{28 \operatorname{atan}{\left(\frac{4}{3} \right)}}{3} + \frac{5 \log{\left(25 \right)}}{2} + \frac{28 \operatorname{atan}{\left(\frac{5}{3} \right)}}{3}$$
-28*atan(4/3)/3 - 5*log(34)/2 + 5*log(25)/2 + 28*atan(5/3)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.