Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)^(3)/(1+cos(x))+sin(x)^(3)/(1-x)
¿Cómo vas a descomponer esta sin(x)^(3)/(1+cos(x))+sin(x)^(3)/(1-x) expresión en fracciones?
Solución
Ha introducido
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3 3 sin (x) sin (x) ---------- + ------- 1 + cos(x) 1 - x
$$\frac{\sin^{3}{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - x}$$
sin(x)^3/(1 + cos(x)) + sin(x)^3/(1 - x)
Simplificación general
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3 sin (x)*(-2 + x - cos(x)) ------------------------- (1 + cos(x))*(-1 + x)
$$\frac{\left(x - \cos{\left(x \right)} - 2\right) \sin^{3}{\left(x \right)}}{\left(x - 1\right) \left(\cos{\left(x \right)} + 1\right)}$$
sin(x)^3*(-2 + x - cos(x))/((1 + cos(x))*(-1 + x))
Denominador racional
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3 3
sin (x)*(1 - x) + sin (x)*(1 + cos(x))
--------------------------------------
(1 - x)*(1 + cos(x))
$$\frac{\left(1 - x\right) \sin^{3}{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin^{3}{\left(x \right)}}{\left(1 - x\right) \left(\cos{\left(x \right)} + 1\right)}$$
(sin(x)^3*(1 - x) + sin(x)^3*(1 + cos(x)))/((1 - x)*(1 + cos(x)))
Unión de expresiones racionales
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3 sin (x)*(2 - x + cos(x)) ------------------------ (1 - x)*(1 + cos(x))
$$\frac{\left(- x + \cos{\left(x \right)} + 2\right) \sin^{3}{\left(x \right)}}{\left(1 - x\right) \left(\cos{\left(x \right)} + 1\right)}$$
sin(x)^3*(2 - x + cos(x))/((1 - x)*(1 + cos(x)))
Potencias
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3 3
/ -I*x I*x\ / -I*x I*x\
I*\- e + e / I*\- e + e /
------------------- + --------------------
8*(1 - x) / I*x -I*x\
| e e |
8*|1 + ---- + -----|
\ 2 2 /
$$\frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}\right)} + \frac{i \left(e^{i x} - e^{- i x}\right)^{3}}{8 \left(1 - x\right)}$$
i*(-exp(-i*x) + exp(i*x))^3/(8*(1 - x)) + i*(-exp(-i*x) + exp(i*x))^3/(8*(1 + exp(i*x)/2 + exp(-i*x)/2))
Combinatoria
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3 sin (x)*(-2 + x - cos(x)) ------------------------- (1 + cos(x))*(-1 + x)
$$\frac{\left(x - \cos{\left(x \right)} - 2\right) \sin^{3}{\left(x \right)}}{\left(x - 1\right) \left(\cos{\left(x \right)} + 1\right)}$$
sin(x)^3*(-2 + x - cos(x))/((1 + cos(x))*(-1 + x))
Parte trigonométrica
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-sin(3*x) + 3*sin(x) -sin(3*x) + 3*sin(x)
-------------------- + --------------------
4*(1 - x) / / pi\\
4*|1 + sin|x + --||
\ \ 2 //
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\sin{\left(x + \frac{\pi}{2} \right)} + 1\right)} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(1 - x\right)}$$
/3*x\ /x\ /3*x\ /x\
2*tan|---| 6*tan|-| 2*tan|---| 6*tan|-|
\ 2 / \2/ \ 2 / \2/
- ------------- + ----------- - ------------- + -----------
2/3*x\ 2/x\ 2/3*x\ 2/x\
1 + tan |---| 1 + tan |-| 1 + tan |---| 1 + tan |-|
\ 2 / \2/ \ 2 / \2/
----------------------------- + -----------------------------
4*(1 - x) / 2/x\\
| 1 - tan |-||
| \2/|
4*|1 + -----------|
| 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} + \frac{- \frac{2 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(1 - x\right)}$$
3/x\ 3/x\
8*cot |-| 8*cot |-|
\2/ \2/
---------------------- + ---------------------------------
3 / 2/x\\
/ 2/x\\ 3 | -1 + cot |-||
|1 + cot |-|| *(1 - x) / 2/x\\ | \2/|
\ \2// |1 + cot |-|| *|1 + ------------|
\ \2// | 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \cot^{3}{\left(\frac{x}{2} \right)}}{\left(1 - x\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
1 1 -------------------- + --------------- / 1 \ 3 3 |1 + ------|*csc (x) (1 - x)*csc (x) \ sec(x)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - x\right) \csc^{3}{\left(x \right)}}$$
1 1 ------------------------- + --------------- / 1 \ 3 3 |1 + -----------|*csc (x) (1 - x)*csc (x) | /pi \| | csc|-- - x|| \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc^{3}{\left(x \right)}} + \frac{1}{\left(1 - x\right) \csc^{3}{\left(x \right)}}$$
/ pi\ / pi\ / pi\ / pi\
- cos|3*x - --| + 3*cos|x - --| - cos|3*x - --| + 3*cos|x - --|
\ 2 / \ 2 / \ 2 / \ 2 /
------------------------------- + -------------------------------
4*(1 - x) 4*(1 + cos(x))
$$\frac{3 \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(3 x - \frac{\pi}{2} \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \cos{\left(x - \frac{\pi}{2} \right)} - \cos{\left(3 x - \frac{\pi}{2} \right)}}{4 \left(1 - x\right)}$$
1 1 ------------------------- + -------------------- / 1 \ 3/ pi\ 3/ pi\ |1 + ------|*sec |x - --| (1 - x)*sec |x - --| \ sec(x)/ \ 2 / \ 2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - x\right) \sec^{3}{\left(x - \frac{\pi}{2} \right)}}$$
-sin(3*x) + 3*sin(x) -sin(3*x) + 3*sin(x)
-------------------- + --------------------
4*(1 - x) 4*(1 + cos(x))
$$\frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(\cos{\left(x \right)} + 1\right)} + \frac{3 \sin{\left(x \right)} - \sin{\left(3 x \right)}}{4 \left(1 - x\right)}$$
3/ pi\ 3/ pi\
cos |x - --| cos |x - --|
\ 2 / \ 2 /
------------ + ------------
1 - x 1 + cos(x)
$$\frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1} + \frac{\cos^{3}{\left(x - \frac{\pi}{2} \right)}}{1 - x}$$
1 3 1 3
- ------------- + ----------- - ------------- + -----------
/ pi\ / pi\ / pi\ / pi\
sec|3*x - --| sec|x - --| sec|3*x - --| sec|x - --|
\ 2 / \ 2 / \ 2 / \ 2 /
----------------------------- + -----------------------------
/ 1 \ 4*(1 - x)
4*|1 + ------|
\ sec(x)/
$$\frac{- \frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}}{4 \left(1 + \frac{1}{\sec{\left(x \right)}}\right)} + \frac{- \frac{1}{\sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(x - \frac{\pi}{2} \right)}}}{4 \left(1 - x\right)}$$
/3*x\ /x\ /3*x\ /x\
2*cot|---| 6*cot|-| 2*cot|---| 6*cot|-|
\ 2 / \2/ \ 2 / \2/
- ------------- + ----------- - ------------- + -----------
2/3*x\ 2/x\ 2/3*x\ 2/x\
1 + cot |---| 1 + cot |-| 1 + cot |---| 1 + cot |-|
\ 2 / \2/ \ 2 / \2/
----------------------------- + -----------------------------
4*(1 - x) / 2/x\\
| -1 + cot |-||
| \2/|
4*|1 + ------------|
| 2/x\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right)} + \frac{- \frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}}{4 \left(1 - x\right)}$$
1 3 1 3 - -------- + ------ - -------- + ------ csc(3*x) csc(x) csc(3*x) csc(x) ------------------- + ------------------- / 1 \ 4*(1 - x) 4*|1 + -----------| | /pi \| | csc|-- - x|| \ \2 //
$$\frac{- \frac{1}{\csc{\left(3 x \right)}} + \frac{3}{\csc{\left(x \right)}}}{4 \left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)} + \frac{- \frac{1}{\csc{\left(3 x \right)}} + \frac{3}{\csc{\left(x \right)}}}{4 \left(1 - x\right)}$$
3/x\ 3/x\
8*tan |-| 8*tan |-|
\2/ \2/
---------------------- + --------------------------------
3 / 2/x\\
/ 2/x\\ 3 | 1 - tan |-||
|1 + tan |-|| *(1 - x) / 2/x\\ | \2/|
\ \2// |1 + tan |-|| *|1 + -----------|
\ \2// | 2/x\|
| 1 + tan |-||
\ \2//
$$\frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{\left(1 - x\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{3}}$$
3 3
sin (x) sin (x)
------- + ---------------
1 - x / pi\
1 + sin|x + --|
\ 2 /
$$\frac{\sin^{3}{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1} + \frac{\sin^{3}{\left(x \right)}}{1 - x}$$
sin(x)^3/(1 - x) + sin(x)^3/(1 + sin(x + pi/2))
Denominador común
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3 3 3
- 2*sin (x) + x*sin (x) - sin (x)*cos(x)
----------------------------------------
-1 + x - cos(x) + x*cos(x)
$$\frac{x \sin^{3}{\left(x \right)} - \sin^{3}{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{3}{\left(x \right)}}{x \cos{\left(x \right)} + x - \cos{\left(x \right)} - 1}$$
(-2*sin(x)^3 + x*sin(x)^3 - sin(x)^3*cos(x))/(-1 + x - cos(x) + x*cos(x))
Respuesta numérica
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sin(x)^3/(1.0 - x) + sin(x)^3/(1.0 + cos(x))
sin(x)^3/(1.0 - x) + sin(x)^3/(1.0 + cos(x))