Sr Examen
Integral de 1/(a*b-a*x-b*x+x^2) dx
Solución
Ha introducido
[src]
x / | | 1 | -------------------- dx | 2 | a*b - a*x - b*x + x | / 0
$$\int\limits_{0}^{x} \frac{1}{x^{2} + \left(- b x + \left(a b - a x\right)\right)}\, dx$$
Integral(1/(a*b - a*x - b*x + x^2), (x, 0, x))
Respuesta (Indefinida)
[src]
/ | | 1 log(-2*x + 2*a) log(-2*b + 2*x) | -------------------- dx = C + --------------- - --------------- | 2 a - b a - b | a*b - a*x - b*x + x | /
$$\int \frac{1}{x^{2} + \left(- b x + \left(a b - a x\right)\right)}\, dx = C + \frac{\log{\left(2 a - 2 x \right)}}{a - b} - \frac{\log{\left(- 2 b + 2 x \right)}}{a - b}$$
Respuesta
[src]
/ 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \
| a b a b a*b | | a b a b a*b | | a b a b a*b | | a b a b a*b |
log|- - - - + --------- + --------- - -----| log|x - - - - - --------- - --------- + -----| log|- - - - - --------- - --------- + -----| log|x - - - - + --------- + --------- - -----|
\ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/
-------------------------------------------- + ---------------------------------------------- - -------------------------------------------- - ----------------------------------------------
a - b a - b a - b a - b
$$- \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} - \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b}$$
=
=
/ 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \
| a b a b a*b | | a b a b a*b | | a b a b a*b | | a b a b a*b |
log|- - - - + --------- + --------- - -----| log|x - - - - - --------- - --------- + -----| log|- - - - - --------- - --------- + -----| log|x - - - - + --------- + --------- - -----|
\ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/ \ 2 2 2*(a - b) 2*(a - b) a - b/
-------------------------------------------- + ---------------------------------------------- - -------------------------------------------- - ----------------------------------------------
a - b a - b a - b a - b
$$- \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} \right)}}{a - b} + \frac{\log{\left(- \frac{a^{2}}{2 \left(a - b\right)} + \frac{a b}{a - b} - \frac{a}{2} - \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b} - \frac{\log{\left(\frac{a^{2}}{2 \left(a - b\right)} - \frac{a b}{a - b} - \frac{a}{2} + \frac{b^{2}}{2 \left(a - b\right)} - \frac{b}{2} + x \right)}}{a - b}$$
log(-a/2 - b/2 + a^2/(2*(a - b)) + b^2/(2*(a - b)) - a*b/(a - b))/(a - b) + log(x - a/2 - b/2 - a^2/(2*(a - b)) - b^2/(2*(a - b)) + a*b/(a - b))/(a - b) - log(-a/2 - b/2 - a^2/(2*(a - b)) - b^2/(2*(a - b)) + a*b/(a - b))/(a - b) - log(x - a/2 - b/2 + a^2/(2*(a - b)) + b^2/(2*(a - b)) - a*b/(a - b))/(a - b)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.