Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sqrt(x^3*cos(x))*(-x^3*sin(x)/2+3*x^2*cos(x)/2)/(x^3*cos(x))
¿Cómo vas a descomponer esta sqrt(x^3*cos(x))*(-x^3*sin(x)/2+3*x^2*cos(x)/2)/(x^3*cos(x)) expresión en fracciones?
Solución
Ha introducido
[src]
___________ / 3 2 \
/ 3 |-x *sin(x) 3*x *cos(x)|
\/ x *cos(x) *|---------- + -----------|
\ 2 2 /
-----------------------------------------
3
x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(\frac{3 x^{2} \cos{\left(x \right)}}{2} + \frac{- x^{3} \sin{\left(x \right)}}{2}\right)}{x^{3} \cos{\left(x \right)}}$$
(sqrt(x^3*cos(x))*(((-x^3)*sin(x))/2 + ((3*x^2)*cos(x))/2))/((x^3*cos(x)))
Simplificación general
[src]
___________
/ 3
\/ x *cos(x) *(3 - x*tan(x))
-----------------------------
2*x
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- x \tan{\left(x \right)} + 3\right)}{2 x}$$
sqrt(x^3*cos(x))*(3 - x*tan(x))/(2*x)
Respuesta numérica
[src]
(x^3*cos(x))^0.5*(1.5*x^2*cos(x) - 0.5*x^3*sin(x))/(x^3*cos(x))
(x^3*cos(x))^0.5*(1.5*x^2*cos(x) - 0.5*x^3*sin(x))/(x^3*cos(x))
Denominador racional
[src]
___________
/ 3 / 3 2 \
\/ x *cos(x) *\- x *sin(x) + 3*x *cos(x)/
------------------------------------------
3
2*x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- x^{3} \sin{\left(x \right)} + 3 x^{2} \cos{\left(x \right)}\right)}{2 x^{3} \cos{\left(x \right)}}$$
sqrt(x^3*cos(x))*(-x^3*sin(x) + 3*x^2*cos(x))/(2*x^3*cos(x))
Potencias
[src]
___________ / 3 2 \
/ 3 | x *sin(x) 3*x *cos(x)|
\/ x *cos(x) *|- --------- + -----------|
\ 2 2 /
------------------------------------------
3
x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- \frac{x^{3} \sin{\left(x \right)}}{2} + \frac{3 x^{2} \cos{\left(x \right)}}{2}\right)}{x^{3} \cos{\left(x \right)}}$$
/ / I*x -I*x\ \
___________________ | 2 |e e | |
/ / I*x -I*x\ |3*x *|---- + -----| 3 / -I*x I*x\|
/ 3 |e e | | \ 2 2 / I*x *\- e + e /|
/ x *|---- + -----| *|------------------- + ---------------------|
\/ \ 2 2 / \ 2 4 /
----------------------------------------------------------------------
/ I*x -I*x\
3 |e e |
x *|---- + -----|
\ 2 2 /
$$\frac{\sqrt{x^{3} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} \left(\frac{i x^{3} \left(e^{i x} - e^{- i x}\right)}{4} + \frac{3 x^{2} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}{2}\right)}{x^{3} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)}$$
sqrt(x^3*(exp(i*x)/2 + exp(-i*x)/2))*(3*x^2*(exp(i*x)/2 + exp(-i*x)/2)/2 + i*x^3*(-exp(-i*x) + exp(i*x))/4)/(x^3*(exp(i*x)/2 + exp(-i*x)/2))
Denominador común
[src]
/ ___________ ___________ \
| / 3 / 3 |
-\- 3*\/ x *cos(x) *cos(x) + x*\/ x *cos(x) *sin(x)/
-------------------------------------------------------
2*x*cos(x)
$$- \frac{x \sqrt{x^{3} \cos{\left(x \right)}} \sin{\left(x \right)} - 3 \sqrt{x^{3} \cos{\left(x \right)}} \cos{\left(x \right)}}{2 x \cos{\left(x \right)}}$$
-(-3*sqrt(x^3*cos(x))*cos(x) + x*sqrt(x^3*cos(x))*sin(x))/(2*x*cos(x))
Combinatoria
[src]
___________
/ 3
-\/ x *cos(x) *(-3*cos(x) + x*sin(x))
---------------------------------------
2*x*cos(x)
$$- \frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right)}{2 x \cos{\left(x \right)}}$$
-sqrt(x^3*cos(x))*(-3*cos(x) + x*sin(x))/(2*x*cos(x))
Unión de expresiones racionales
[src]
___________
/ 3
\/ x *cos(x) *(3*cos(x) - x*sin(x))
------------------------------------
2*x*cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- x \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right)}{2 x \cos{\left(x \right)}}$$
sqrt(x^3*cos(x))*(3*cos(x) - x*sin(x))/(2*x*cos(x))
Compilar la expresión
[src]
___________ / 3 2 \
/ 3 | x *sin(x) 3*x *cos(x)|
\/ x *cos(x) *|- --------- + -----------|
\ 2 2 /
------------------------------------------
3
x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- \frac{x^{3} \sin{\left(x \right)}}{2} + \frac{3 x^{2} \cos{\left(x \right)}}{2}\right)}{x^{3} \cos{\left(x \right)}}$$
sqrt(x^3*cos(x))*(-x^3*sin(x)/2 + 3*x^2*cos(x)/2)/(x^3*cos(x))
Parte trigonométrica
[src]
___________________
/ 3 / 2/x\\ / 3 /x\ 2 / 2/x\\\
/ x *|-1 + cot |-|| | x *cot|-| 3*x *|-1 + cot |-|||
/ \ \2// / 2/x\\ | \2/ \ \2//|
/ ----------------- *|1 + cot |-||*|- ----------- + -------------------|
/ 2/x\ \ \2// | 2/x\ / 2/x\\ |
/ 1 + cot |-| | 1 + cot |-| 2*|1 + cot |-|| |
\/ \2/ \ \2/ \ \2// /
-------------------------------------------------------------------------------
3 / 2/x\\
x *|-1 + cot |-||
\ \2//
$$\frac{\sqrt{\frac{x^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- \frac{x^{3} \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 x^{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}{x^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}$$
/ / pi\\
___________ | x*cos|x - --||
/ 3 | \ 2 /|
\/ x *cos(x) *|3 - -------------|
\ cos(x) /
----------------------------------
2*x
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- \frac{x \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + 3\right)}{2 x}$$
_____________ / /pi \\
/ 3 | x*csc|-- - x||
/ x | \2 /|
/ ----------- *|3 - -------------|
/ /pi \ \ csc(x) /
/ csc|-- - x|
\/ \2 /
----------------------------------------
2*x
$$\frac{\sqrt{\frac{x^{3}}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \left(- \frac{x \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + 3\right)}{2 x}$$
________
/ 3 / 3 2 \
/ x | x 3*x |
/ ------ *|- -------- + --------|*sec(x)
\/ sec(x) \ 2*csc(x) 2*sec(x)/
--------------------------------------------
3
x
$$\frac{\sqrt{\frac{x^{3}}{\sec{\left(x \right)}}} \left(- \frac{x^{3}}{2 \csc{\left(x \right)}} + \frac{3 x^{2}}{2 \sec{\left(x \right)}}\right) \sec{\left(x \right)}}{x^{3}}$$
___________ / 3 2 \
/ 3 | x *sin(x) 3*x *cos(x)|
\/ x *cos(x) *|- --------- + -----------|
\ 2 2 /
------------------------------------------
3
x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- \frac{x^{3} \sin{\left(x \right)}}{2} + \frac{3 x^{2} \cos{\left(x \right)}}{2}\right)}{x^{3} \cos{\left(x \right)}}$$
__________________
/ 3 / 2/x\\ / 3 /x\ 2 / 2/x\\\
/ x *|1 - tan |-|| | x *tan|-| 3*x *|1 - tan |-|||
/ \ \2// / 2/x\\ | \2/ \ \2//|
/ ---------------- *|1 + tan |-||*|- ----------- + ------------------|
/ 2/x\ \ \2// | 2/x\ / 2/x\\ |
/ 1 + tan |-| | 1 + tan |-| 2*|1 + tan |-|| |
\/ \2/ \ \2/ \ \2// /
-----------------------------------------------------------------------------
3 / 2/x\\
x *|1 - tan |-||
\ \2//
$$\frac{\sqrt{\frac{x^{3} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- \frac{x^{3} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{3 x^{2} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{x^{3} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}$$
2
x *(3*cos(x) - x*sin(x))
------------------------
___________
/ 3
2*\/ x *cos(x)
$$\frac{x^{2} \left(- x \sin{\left(x \right)} + 3 \cos{\left(x \right)}\right)}{2 \sqrt{x^{3} \cos{\left(x \right)}}}$$
/ 3 / pi\ \
___________ | x *cos|x - --| 2 |
/ 3 | \ 2 / 3*x *cos(x)|
\/ x *cos(x) *|- -------------- + -----------|
\ 2 2 /
-----------------------------------------------
3
x *cos(x)
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- \frac{x^{3} \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{3 x^{2} \cos{\left(x \right)}}{2}\right)}{x^{3} \cos{\left(x \right)}}$$
_____________
/ 3
/ x / 3 x \
/ ----------- *|- ----------- + ------|
/ /pi \ | /pi \ csc(x)|
/ csc|-- - x| | csc|-- - x| |
\/ \2 / \ \2 / /
---------------------------------------------
/ 1 1 \
2*x*|------- - ------------|
| 2/x\ 2/pi x\|
|csc |-| csc |-- - -||
\ \2/ \2 2//
$$\frac{\sqrt{\frac{x^{3}}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \left(\frac{x}{\csc{\left(x \right)}} - \frac{3}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)}{2 x \left(- \frac{1}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)}$$
___________
/ 3 / / pi\\
\/ x *cos(x) *|-3*cos(x) + x*cos|x - --||
\ \ 2 //
------------------------------------------
/ 2/x pi\ 2/x\\
2*x*|cos |- - --| - cos |-||
\ \2 2 / \2//
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(x \cos{\left(x - \frac{\pi}{2} \right)} - 3 \cos{\left(x \right)}\right)}{2 x \left(- \cos^{2}{\left(\frac{x}{2} \right)} + \cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}\right)}$$
________________
/ 3 / pi\ / / pi\ \
/ x *sin|x + --| *|- 3*sin|x + --| + x*sin(x)|
\/ \ 2 / \ \ 2 / /
-------------------------------------------------
/ 2/x\ 2/pi x\\
2*x*|sin |-| - sin |-- + -||
\ \2/ \2 2//
$$\frac{\sqrt{x^{3} \sin{\left(x + \frac{\pi}{2} \right)}} \left(x \sin{\left(x \right)} - 3 \sin{\left(x + \frac{\pi}{2} \right)}\right)}{2 x \left(\sin^{2}{\left(\frac{x}{2} \right)} - \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}\right)}$$
_____________
/ 3 / 3 2 \
/ x | x 3*x | /pi \
/ ----------- *|- -------- + -------------|*csc|-- - x|
/ /pi \ | 2*csc(x) /pi \| \2 /
/ csc|-- - x| | 2*csc|-- - x||
\/ \2 / \ \2 //
-------------------------------------------------------------
3
x
$$\frac{\sqrt{\frac{x^{3}}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \left(- \frac{x^{3}}{2 \csc{\left(x \right)}} + \frac{3 x^{2}}{2 \csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}{x^{3}}$$
__________________
/ 3 / 2/x\\
/ x *|1 - tan |-||
/ \ \2//
/ ---------------- *(3 - x*tan(x))
/ 2/x\
/ 1 + tan |-|
\/ \2/
-----------------------------------------
2*x
$$\frac{\sqrt{\frac{x^{3} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- x \tan{\left(x \right)} + 3\right)}{2 x}$$
___________
/ 3
\/ x *cos(x) *(-3*cos(x) + x*sin(x))
-------------------------------------
/ 2/x\ 2/x\\
2*x*|sin |-| - cos |-||
\ \2/ \2//
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(x \sin{\left(x \right)} - 3 \cos{\left(x \right)}\right)}{2 x \left(\sin^{2}{\left(\frac{x}{2} \right)} - \cos^{2}{\left(\frac{x}{2} \right)}\right)}$$
________________ / 2 \
/ 3 / pi\ | 2*x*sin (x)|
/ x *sin|x + --| *|3 - -----------|
\/ \ 2 / \ sin(2*x) /
--------------------------------------
2*x
$$\frac{\sqrt{x^{3} \sin{\left(x + \frac{\pi}{2} \right)}} \left(- \frac{2 x \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 3\right)}{2 x}$$
___________
/ 3
-\/ x *cos(x) *(-3 + x*tan(x))
--------------------------------
2*x
$$- \frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(x \tan{\left(x \right)} - 3\right)}{2 x}$$
________
/ 3
/ x / 3 x \
/ ------ *|- ------ + -----------|
\/ sec(x) | sec(x) / pi\|
| sec|x - --||
\ \ 2 //
--------------------------------------
/ 1 1 \
2*x*|------------ - -------|
| 2/x pi\ 2/x\|
|sec |- - --| sec |-||
\ \2 2 / \2//
$$\frac{\sqrt{\frac{x^{3}}{\sec{\left(x \right)}}} \left(\frac{x}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{3}{\sec{\left(x \right)}}\right)}{2 x \left(\frac{1}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(\frac{x}{2} \right)}}\right)}$$
________
/ 3 / 3 2 \
/ x | x 3*x |
/ ------ *|- ------------- + --------|*sec(x)
\/ sec(x) | / pi\ 2*sec(x)|
| 2*sec|x - --| |
\ \ 2 / /
-------------------------------------------------
3
x
$$\frac{\sqrt{\frac{x^{3}}{\sec{\left(x \right)}}} \left(- \frac{x^{3}}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{3 x^{2}}{2 \sec{\left(x \right)}}\right) \sec{\left(x \right)}}{x^{3}}$$
________
/ 3
/ x / x*sec(x) \
/ ------ *|3 - -----------|
\/ sec(x) | / pi\|
| sec|x - --||
\ \ 2 //
-------------------------------
2*x
$$\frac{\sqrt{\frac{x^{3}}{\sec{\left(x \right)}}} \left(- \frac{x \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 3\right)}{2 x}$$
/ 2 / pi\\
________________ | 3 3*x *sin|x + --||
/ 3 / pi\ | x *sin(x) \ 2 /|
/ x *sin|x + --| *|- --------- + ----------------|
\/ \ 2 / \ 2 2 /
-----------------------------------------------------
3 / pi\
x *sin|x + --|
\ 2 /
$$\frac{\sqrt{x^{3} \sin{\left(x + \frac{\pi}{2} \right)}} \left(- \frac{x^{3} \sin{\left(x \right)}}{2} + \frac{3 x^{2} \sin{\left(x + \frac{\pi}{2} \right)}}{2}\right)}{x^{3} \sin{\left(x + \frac{\pi}{2} \right)}}$$
__________________
/ 3 / 2/x\\ / / 2/x\\ /x\\
/ x *|1 - tan |-|| | 3*|1 - tan |-|| 2*x*tan|-||
/ \ \2// | \ \2// \2/|
/ ---------------- *|- --------------- + -----------|
/ 2/x\ | 2/x\ 2/x\|
/ 1 + tan |-| | 1 + tan |-| 1 + tan |-||
\/ \2/ \ \2/ \2//
------------------------------------------------------------
/ 2 \
| / 2/x\\ 2/x\ |
| |1 - tan |-|| 4*tan |-| |
| \ \4// \4/ |
2*x*|- -------------- + --------------|
| 2 2|
| / 2/x\\ / 2/x\\ |
| |1 + tan |-|| |1 + tan |-|| |
\ \ \4// \ \4// /
$$\frac{\sqrt{\frac{x^{3} \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{3 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{2 x \left(- \frac{\left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{x}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}$$
___________________
/ 3 / 2/x\\ / / 2/x\\ /x\\
/ x *|-1 + cot |-|| | 3*|-1 + cot |-|| 2*x*cot|-||
/ \ \2// | \ \2// \2/|
/ ----------------- *|- ---------------- + -----------|
/ 2/x\ | 2/x\ 2/x\|
/ 1 + cot |-| | 1 + cot |-| 1 + cot |-||
\/ \2/ \ \2/ \2//
--------------------------------------------------------------
/ 2 \
| / 2/x\\ 2/x\ |
| |-1 + cot |-|| 4*cot |-| |
| \ \4// \4/ |
2*x*|- --------------- + --------------|
| 2 2|
| / 2/x\\ / 2/x\\ |
| |1 + cot |-|| |1 + cot |-|| |
\ \ \4// \ \4// /
$$\frac{\sqrt{\frac{x^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\frac{2 x \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{3 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)}{2 x \left(- \frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{x}{4} \right)}}{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{2}}\right)}$$
___________
/ 3
\/ x *cos(x) *(3 - x*tan(x))
-----------------------------
2*x
$$\frac{\sqrt{x^{3} \cos{\left(x \right)}} \left(- x \tan{\left(x \right)} + 3\right)}{2 x}$$
___________________
/ 3 / 2/x\\
/ x *|-1 + cot |-||
/ \ \2// / x \
/ ----------------- *|3 - ------|
/ 2/x\ \ cot(x)/
/ 1 + cot |-|
\/ \2/
----------------------------------------
2*x
$$\frac{\sqrt{\frac{x^{3} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(- \frac{x}{\cot{\left(x \right)}} + 3\right)}{2 x}$$
sqrt(x^3*(-1 + cot(x/2)^2)/(1 + cot(x/2)^2))*(3 - x/cot(x))/(2*x)