Sr Examen
Integral de a*x/(b*x+c*x^2) dx
Solución
Ha introducido
[src]
1 / | | a*x | ---------- dx | 2 | b*x + c*x | / 0
$$\int\limits_{0}^{1} \frac{a x}{b x + c x^{2}}\, dx$$
Integral((a*x)/(b*x + c*x^2), (x, 0, 1))
Solución detallada
Tenemos el integral:
/ | | a*x | ---------- dx | 2 | b*x + c*x | /
Reescribimos la función subintegral
a*x a 2*c*x + b
---------- = ---*----------
2 2*c 2
b*x + c*x c*x + b*xo
/ | | a*x | ---------- dx | 2 = | b*x + c*x | /
/
|
| 2*c*x + b
a* | ---------- dx
| 2
| c*x + b*x
|
/
------------------
2*c En integral
/
|
| 2*c*x + b
a* | ---------- dx
| 2
| c*x + b*x
|
/
------------------
2*c hacemos el cambio
2 u = b*x + c*x
entonces
integral =
/ | | 1 a* | - du | u | / a*log(u) --------- = -------- 2*c 2*c
hacemos cambio inverso
/
|
| 2*c*x + b
a* | ---------- dx
| 2
| c*x + b*x
| / 2\
/ a*log\b*x + c*x /
------------------ = -----------------
2*c 2*c La solución:
a*log(b + c*x)
C + --------------
c
Respuesta (Indefinida)
[src]
/ // b \ // b \\
| || x + --- | || x + --- ||
| || 2*c | || 2*c ||
| || ------- for c = 0| || ------- for c = 0||
| || b | || b ||
| 2*c*|< | 2*c*|< ||
| || / / b \\ | || / / b \\ ||
| ||log|b + 2*c*|x + ---|| | ||-log|b - 2*c*|x + ---|| ||
| || \ \ 2*c// | || \ \ 2*c// ||
| ||---------------------- otherwise| ||------------------------ otherwise||
/ | \\ 2*c / \\ 2*c /|
| / 2\ a*b*|- ---------------------------------------- - ------------------------------------------|
| a*x a*log\b*x + c*x / \ b b /
| ---------- dx = C + ----------------- - ---------------------------------------------------------------------------------------------
| 2 2*c 2*c
| b*x + c*x
|
/
$$\int \frac{a x}{b x + c x^{2}}\, dx = C - \frac{a b \left(- \frac{2 c \left(\begin{cases} \frac{\frac{b}{2 c} + x}{b} & \text{for}\: c = 0 \\- \frac{\log{\left(b - 2 c \left(\frac{b}{2 c} + x\right) \right)}}{2 c} & \text{otherwise} \end{cases}\right)}{b} - \frac{2 c \left(\begin{cases} \frac{\frac{b}{2 c} + x}{b} & \text{for}\: c = 0 \\\frac{\log{\left(b + 2 c \left(\frac{b}{2 c} + x\right) \right)}}{2 c} & \text{otherwise} \end{cases}\right)}{b}\right)}{2 c} + \frac{a \log{\left(b x + c x^{2} \right)}}{2 c}$$
Respuesta
[src]
a*log(b + c) a*log(b)
------------ - --------
c c
$$- \frac{a \log{\left(b \right)}}{c} + \frac{a \log{\left(b + c \right)}}{c}$$
=
=
a*log(b + c) a*log(b)
------------ - --------
c c
$$- \frac{a \log{\left(b \right)}}{c} + \frac{a \log{\left(b + c \right)}}{c}$$
a*log(b + c)/c - a*log(b)/c
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.