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