Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)/2+x/(1+x^2)-x*sin(x)/2+atan(x)
¿Cómo vas a descomponer esta cos(x)/2+x/(1+x^2)-x*sin(x)/2+atan(x) expresión en fracciones?
Solución
Ha introducido
[src]
cos(x) x x*sin(x)
------ + ------ - -------- + atan(x)
2 2 2
1 + x
$$\left(- \frac{x \sin{\left(x \right)}}{2} + \left(\frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2}\right)\right) + \operatorname{atan}{\left(x \right)}$$
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Simplificación general
[src]
/ 2\
\1 + x /*(2*atan(x) - x*sin(x) + cos(x))
x + ----------------------------------------
2
--------------------------------------------
2
1 + x
$$\frac{x + \frac{\left(x^{2} + 1\right) \left(- x \sin{\left(x \right)} + \cos{\left(x \right)} + 2 \operatorname{atan}{\left(x \right)}\right)}{2}}{x^{2} + 1}$$
(x + (1 + x^2)*(2*atan(x) - x*sin(x) + cos(x))/2)/(1 + x^2)
Respuesta numérica
[src]
0.5*cos(x) + x/(1.0 + x^2) - 0.5*x*sin(x) + atan(x)
0.5*cos(x) + x/(1.0 + x^2) - 0.5*x*sin(x) + atan(x)
Denominador racional
[src]
/ 2\ / 2\ / 2\
4*x + \4 + 4*x /*atan(x) + 2*\1 + x /*cos(x) - x*\2 + 2*x /*sin(x)
------------------------------------------------------------------
2
4 + 4*x
$$\frac{- x \left(2 x^{2} + 2\right) \sin{\left(x \right)} + 4 x + 2 \left(x^{2} + 1\right) \cos{\left(x \right)} + \left(4 x^{2} + 4\right) \operatorname{atan}{\left(x \right)}}{4 x^{2} + 4}$$
(4*x + (4 + 4*x^2)*atan(x) + 2*(1 + x^2)*cos(x) - x*(2 + 2*x^2)*sin(x))/(4 + 4*x^2)
Potencias
[src]
I*x -I*x / -I*x I*x\
e e x I*x*\- e + e /
---- + ----- + ------ + -------------------- + atan(x)
4 4 2 4
1 + x
$$\frac{i x \left(e^{i x} - e^{- i x}\right)}{4} + \frac{x}{x^{2} + 1} + \frac{e^{i x}}{4} + \operatorname{atan}{\left(x \right)} + \frac{e^{- i x}}{4}$$
cos(x) x x*sin(x)
------ + ------ - -------- + atan(x)
2 2 2
1 + x
$$- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Compilar la expresión
[src]
cos(x) x x*sin(x)
------ + ------ - -------- + atan(x)
2 2 2
1 + x
$$- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Parte trigonométrica
[src]
1 x x
-------- + ------ - ------------- + atan(x)
2*sec(x) 2 / pi\
1 + x 2*sec|x - --|
\ 2 /
$$- \frac{x}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \sec{\left(x \right)}}$$
/ pi\
x*cos|x - --|
cos(x) x \ 2 /
------ + ------ - ------------- + atan(x)
2 2 2
1 + x
$$- \frac{x \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
1 x x
-------- + ------ - -------- + atan(x)
2*sec(x) 2 2*csc(x)
1 + x
$$- \frac{x}{2 \csc{\left(x \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \sec{\left(x \right)}}$$
2/x\ /x\
1 - tan |-| x*tan|-|
x \2/ \2/
------ + --------------- - ----------- + atan(x)
2 / 2/x\\ 2/x\
1 + x 2*|1 + tan |-|| 1 + tan |-|
\ \2// \2/
$$- \frac{x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{x}{x^{2} + 1} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \operatorname{atan}{\left(x \right)}$$
2/x\ /x\
-1 + cot |-| x*cot|-|
x \2/ \2/
------ + --------------- - ----------- + atan(x)
2 / 2/x\\ 2/x\
1 + x 2*|1 + cot |-|| 1 + cot |-|
\ \2// \2/
$$- \frac{x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{x}{x^{2} + 1} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \operatorname{atan}{\left(x \right)}$$
cos(x) x x*sin(x)
------ + ------ - -------- + atan(x)
2 2 2
1 + x
$$- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
1 x x
------------- + ------ - -------- + atan(x)
/pi \ 2 2*csc(x)
2*csc|-- - x| 1 + x
\2 /
$$- \frac{x}{2 \csc{\left(x \right)}} + \frac{x}{x^{2} + 1} + \operatorname{atan}{\left(x \right)} + \frac{1}{2 \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ pi\
sin|x + --|
\ 2 / x x*sin(x)
----------- + ------ - -------- + atan(x)
2 2 2
1 + x
$$- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
sin(x + pi/2)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Denominador común
[src]
cos(x) x x*sin(x)
------ + ------ - -------- + atan(x)
2 2 2
1 + x
$$- \frac{x \sin{\left(x \right)}}{2} + \frac{x}{x^{2} + 1} + \frac{\cos{\left(x \right)}}{2} + \operatorname{atan}{\left(x \right)}$$
cos(x)/2 + x/(1 + x^2) - x*sin(x)/2 + atan(x)
Unión de expresiones racionales
[src]
/ 2\ / 2\ / 2\
2*x + \1 + x /*cos(x) + 2*\1 + x /*atan(x) - x*\1 + x /*sin(x)
--------------------------------------------------------------
/ 2\
2*\1 + x /
$$\frac{- x \left(x^{2} + 1\right) \sin{\left(x \right)} + 2 x + \left(x^{2} + 1\right) \cos{\left(x \right)} + 2 \left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}$$
(2*x + (1 + x^2)*cos(x) + 2*(1 + x^2)*atan(x) - x*(1 + x^2)*sin(x))/(2*(1 + x^2))
Combinatoria
[src]
/ 3 2 2 \
-\-cos(x) - 2*x - 2*atan(x) + x*sin(x) + x *sin(x) - x *cos(x) - 2*x *atan(x)/
-------------------------------------------------------------------------------
/ 2\
2*\1 + x /
$$- \frac{x^{3} \sin{\left(x \right)} - x^{2} \cos{\left(x \right)} - 2 x^{2} \operatorname{atan}{\left(x \right)} + x \sin{\left(x \right)} - 2 x - \cos{\left(x \right)} - 2 \operatorname{atan}{\left(x \right)}}{2 \left(x^{2} + 1\right)}$$
-(-cos(x) - 2*x - 2*atan(x) + x*sin(x) + x^3*sin(x) - x^2*cos(x) - 2*x^2*atan(x))/(2*(1 + x^2))