Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(x^2+x+1)/9+(2*atan((2*x+1)/sqrt(3)))/3^(3/2)-(2*log(x-1))/9-x/(3*x^3-3)
¿Cómo vas a descomponer esta log(x^2+x+1)/9+(2*atan((2*x+1)/sqrt(3)))/3^(3/2)-(2*log(x-1))/9-x/(3*x^3-3) expresión en fracciones?
Solución
Ha introducido
[src]
/2*x + 1\
2*atan|-------|
/ 2 \ | ___ |
log\x + x + 1/ \ \/ 3 / 2*log(x - 1) x
--------------- + --------------- - ------------ - --------
9 3/2 9 3
3 3*x - 3
$$- \frac{x}{3 x^{3} - 3} + \left(\left(\frac{\log{\left(\left(x^{2} + x\right) + 1 \right)}}{9} + \frac{2 \operatorname{atan}{\left(\frac{2 x + 1}{\sqrt{3}} \right)}}{3^{\frac{3}{2}}}\right) - \frac{2 \log{\left(x - 1 \right)}}{9}\right)$$
log(x^2 + x + 1)/9 + (2*atan((2*x + 1)/sqrt(3)))/3^(3/2) - 2*log(x - 1)/9 - x/(3*x^3 - 3)
Simplificación general
[src]
/ / ___ \ \
/ 3\ | ___ |\/ 3 *(1 + 2*x)| / 2\|
-3*x + \-1 + x /*|-2*log(-1 + x) + 2*\/ 3 *atan|---------------| + log\1 + x + x /|
\ \ 3 / /
-----------------------------------------------------------------------------------
/ 3\
9*\-1 + x /
$$\frac{- 3 x + \left(x^{3} - 1\right) \left(- 2 \log{\left(x - 1 \right)} + \log{\left(x^{2} + x + 1 \right)} + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}\right)}{9 \left(x^{3} - 1\right)}$$
(-3*x + (-1 + x^3)*(-2*log(-1 + x) + 2*sqrt(3)*atan(sqrt(3)*(1 + 2*x)/3) + log(1 + x + x^2)))/(9*(-1 + x^3))
Unión de expresiones racionales
[src]
/ / ___ \ \
/ 3\ | ___ |\/ 3 *(1 + 2*x)| |
-3*x + \-1 + x /*|-2*log(-1 + x) + 2*\/ 3 *atan|---------------| + log(1 + x*(1 + x))|
\ \ 3 / /
--------------------------------------------------------------------------------------
/ 3\
9*\-1 + x /
$$\frac{- 3 x + \left(x^{3} - 1\right) \left(- 2 \log{\left(x - 1 \right)} + \log{\left(x \left(x + 1\right) + 1 \right)} + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}\right)}{9 \left(x^{3} - 1\right)}$$
(-3*x + (-1 + x^3)*(-2*log(-1 + x) + 2*sqrt(3)*atan(sqrt(3)*(1 + 2*x)/3) + log(1 + x*(1 + x))))/(9*(-1 + x^3))
Potencias
[src]
/ ___ \
___ |\/ 3 *(1 + 2*x)|
/ 2\ 2*\/ 3 *atan|---------------|
2*log(-1 + x) log\1 + x + x / x \ 3 /
- ------------- + --------------- - --------- + -----------------------------
9 9 3 9
-3 + 3*x
$$- \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left(x - 1 \right)}}{9} + \frac{\log{\left(x^{2} + x + 1 \right)}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}}{9}$$
___ / ___ /1 2*x\\
/ 2\ 2*\/ 3 *atan|\/ 3 *|- + ---||
2*log(-1 + x) log\1 + x + x / x \ \3 3 //
- ------------- + --------------- - --------- + -----------------------------
9 9 3 9
-3 + 3*x
$$- \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left(x - 1 \right)}}{9} + \frac{\log{\left(x^{2} + x + 1 \right)}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\sqrt{3} \left(\frac{2 x}{3} + \frac{1}{3}\right) \right)}}{9}$$
-2*log(-1 + x)/9 + log(1 + x + x^2)/9 - x/(-3 + 3*x^3) + 2*sqrt(3)*atan(sqrt(3)*(1/3 + 2*x/3))/9
Compilar la expresión
[src]
___ /2*x + 1\
2*\/ 3 *atan|-------|
/ 2 \ | ___ |
2*log(x - 1) log\x + x + 1/ x \ \/ 3 /
- ------------ + --------------- - --------- + ---------------------
9 9 3 9
-3 + 3*x
$$- \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left(x - 1 \right)}}{9} + \frac{\log{\left(\left(x^{2} + x\right) + 1 \right)}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 x + 1}{\sqrt{3}} \right)}}{9}$$
-2*log(x - 1)/9 + log(x^2 + x + 1)/9 - x/(-3 + 3*x^3) + 2*sqrt(3)*atan((2*x + 1)/sqrt(3))/9
Denominador racional
[src]
/ ___ ___\ / ___ ___\
/ 2\ 3 / 2\ ___ |\/ 3 2*x*\/ 3 | 3 ___ 3 |\/ 3 2*x*\/ 3 |
- log\1 + x + x / - 3*x + 2*log(-1 + x) + x *log\1 + x + x / - 2*\/ 3 *atan|----- + ---------| - 2*x *log(-1 + x) + 2*\/ 3 *x *atan|----- + ---------|
\ 3 3 / \ 3 3 /
------------------------------------------------------------------------------------------------------------------------------------------------------
3
-9 + 9*x
$$\frac{- 2 x^{3} \log{\left(x - 1 \right)} + x^{3} \log{\left(x^{2} + x + 1 \right)} + 2 \sqrt{3} x^{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right)} - 3 x + 2 \log{\left(x - 1 \right)} - \log{\left(x^{2} + x + 1 \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right)}}{9 x^{3} - 9}$$
(-log(1 + x + x^2) - 3*x + 2*log(-1 + x) + x^3*log(1 + x + x^2) - 2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3) - 2*x^3*log(-1 + x) + 2*sqrt(3)*x^3*atan(sqrt(3)/3 + 2*x*sqrt(3)/3))/(-9 + 9*x^3)
Respuesta numérica
[src]
0.111111111111111*log(x^2 + x + 1) + 0.384900179459751*atan((2*x + 1)/sqrt(3)) - 0.222222222222222*log(x - 1) - x/(-3.0 + 3.0*x^3)
0.111111111111111*log(x^2 + x + 1) + 0.384900179459751*atan((2*x + 1)/sqrt(3)) - 0.222222222222222*log(x - 1) - x/(-3.0 + 3.0*x^3)
Combinatoria
[src]
/ ___ ___\ / ___ ___\
/ 2\ 3 / 2\ ___ |\/ 3 2*x*\/ 3 | 3 ___ 3 |\/ 3 2*x*\/ 3 |
- log\1 + x + x / - 3*x + 2*log(-1 + x) + x *log\1 + x + x / - 2*\/ 3 *atan|----- + ---------| - 2*x *log(-1 + x) + 2*\/ 3 *x *atan|----- + ---------|
\ 3 3 / \ 3 3 /
------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2\
9*(-1 + x)*\1 + x + x /
$$\frac{- 2 x^{3} \log{\left(x - 1 \right)} + x^{3} \log{\left(x^{2} + x + 1 \right)} + 2 \sqrt{3} x^{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right)} - 3 x + 2 \log{\left(x - 1 \right)} - \log{\left(x^{2} + x + 1 \right)} - 2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right)}}{9 \left(x - 1\right) \left(x^{2} + x + 1\right)}$$
(-log(1 + x + x^2) - 3*x + 2*log(-1 + x) + x^3*log(1 + x + x^2) - 2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3) - 2*x^3*log(-1 + x) + 2*sqrt(3)*x^3*atan(sqrt(3)/3 + 2*x*sqrt(3)/3))/(9*(-1 + x)*(1 + x + x^2))
Denominador común
[src]
/ ___ ___\
___ |\/ 3 2*x*\/ 3 |
/ 2\ 2*\/ 3 *atan|----- + ---------|
2*log(-1 + x) log\1 + x + x / x \ 3 3 /
- ------------- + --------------- - --------- + -------------------------------
9 9 3 9
-3 + 3*x
$$- \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left(x - 1 \right)}}{9} + \frac{\log{\left(x^{2} + x + 1 \right)}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right)}}{9}$$
-2*log(-1 + x)/9 + log(1 + x + x^2)/9 - x/(-3 + 3*x^3) + 2*sqrt(3)*atan(sqrt(3)/3 + 2*x*sqrt(3)/3)/9
Parte trigonométrica
[src]
/ ___ \
___ |\/ 3 *(1 + 2*x)|
/ 2\ 2*\/ 3 *atan|---------------|
2*log(-1 + x) log\1 + x + x / x \ 3 /
- ------------- + --------------- - --------- + -----------------------------
9 9 3 9
-3 + 3*x
$$- \frac{x}{3 x^{3} - 3} - \frac{2 \log{\left(x - 1 \right)}}{9} + \frac{\log{\left(x^{2} + x + 1 \right)}}{9} + \frac{2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 x + 1\right)}{3} \right)}}{9}$$
-2*log(-1 + x)/9 + log(1 + x + x^2)/9 - x/(-3 + 3*x^3) + 2*sqrt(3)*atan(sqrt(3)*(1 + 2*x)/3)/9