Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cot(x)^2*((2+2*tan(x)^2)*tan(x)/cot(x)^2-tan(x)^2*(-2-2*cot(x)^2)/cot(x)^3)/(2*tan(x)^2)
¿Cómo vas a descomponer esta cot(x)^2*((2+2*tan(x)^2)*tan(x)/cot(x)^2-tan(x)^2*(-2-2*cot(x)^2)/cot(x)^3)/(2*tan(x)^2) expresión en fracciones?
Solución
Ha introducido
[src]
// 2 \ 2 / 2 \\
2 |\2 + 2*tan (x)/*tan(x) tan (x)*\-2 - 2*cot (x)/|
cot (x)*|---------------------- - ------------------------|
| 2 3 |
\ cot (x) cot (x) /
-----------------------------------------------------------
2
2*tan (x)
$$\frac{\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{\cot^{2}{\left(x \right)}} - \frac{\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \tan^{2}{\left(x \right)}}{\cot^{3}{\left(x \right)}}\right) \cot^{2}{\left(x \right)}}{2 \tan^{2}{\left(x \right)}}$$
(cot(x)^2*(((2 + 2*tan(x)^2)*tan(x))/cot(x)^2 - tan(x)^2*(-2 - 2*cot(x)^2)/cot(x)^3))/((2*tan(x)^2))
Respuesta numérica
[src]
0.5*cot(x)^2*((2.0 + 2.0*tan(x)^2)*tan(x)/cot(x)^2 - tan(x)^2*(-2.0 - 2.0*cot(x)^2)/cot(x)^3)/tan(x)^2
0.5*cot(x)^2*((2.0 + 2.0*tan(x)^2)*tan(x)/cot(x)^2 - tan(x)^2*(-2.0 - 2.0*cot(x)^2)/cot(x)^3)/tan(x)^2
Potencias
[src]
/ 2 / 2 \ / 2 \ \
2 |tan (x)*\2 + 2*cot (x)/ \2 + 2*tan (x)/*tan(x)|
cot (x)*|----------------------- + ----------------------|
| 3 2 |
\ 2*cot (x) 2*cot (x) /
----------------------------------------------------------
2
tan (x)
$$\frac{\left(\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)}}{2 \cot^{2}{\left(x \right)}} + \frac{\left(2 \cot^{2}{\left(x \right)} + 2\right) \tan^{2}{\left(x \right)}}{2 \cot^{3}{\left(x \right)}}\right) \cot^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)}}$$
/ / 2\ \
| | / I*x -I*x\ | |
| | 2*\- e + e / | / I*x -I*x\|
| I*|2 - -------------------|*\- e + e /|
| 2 | 2 | |
2 |/ I*x -I*x\ / 2 \ | / I*x -I*x\ | |
/ I*x -I*x\ 2 |\- e + e / *\-2 - 2*cot (x)/ \ \e + e / / |
-\e + e / *cot (x)*|---------------------------------- + --------------------------------------------|
| 2 / I*x -I*x\ 2 |
| / I*x -I*x\ 3 \e + e /*cot (x) |
\ \e + e / *cot (x) /
-------------------------------------------------------------------------------------------------------------
2
/ I*x -I*x\
2*\- e + e /
$$- \frac{\left(\frac{i \left(- \frac{2 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 2\right) \left(- e^{i x} + e^{- i x}\right)}{\left(e^{i x} + e^{- i x}\right) \cot^{2}{\left(x \right)}} + \frac{\left(- e^{i x} + e^{- i x}\right)^{2} \left(- 2 \cot^{2}{\left(x \right)} - 2\right)}{\left(e^{i x} + e^{- i x}\right)^{2} \cot^{3}{\left(x \right)}}\right) \left(e^{i x} + e^{- i x}\right)^{2} \cot^{2}{\left(x \right)}}{2 \left(- e^{i x} + e^{- i x}\right)^{2}}$$
-(exp(i*x) + exp(-i*x))^2*cot(x)^2*((-exp(i*x) + exp(-i*x))^2*(-2 - 2*cot(x)^2)/((exp(i*x) + exp(-i*x))^2*cot(x)^3) + i*(2 - 2*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(-exp(i*x) + exp(-i*x))/((exp(i*x) + exp(-i*x))*cot(x)^2))/(2*(-exp(i*x) + exp(-i*x))^2)
Denominador común
[src]
cot(x) + tan(x) --------------- + cot(x) + tan(x) cot(x)*tan(x)
$$\frac{\tan{\left(x \right)} + \cot{\left(x \right)}}{\tan{\left(x \right)} \cot{\left(x \right)}} + \tan{\left(x \right)} + \cot{\left(x \right)}$$
(cot(x) + tan(x))/(cot(x)*tan(x)) + cot(x) + tan(x)
Denominador racional
[src]
3 / 2 \ 2 2 / 2 \
cot (x)*\2 + 2*tan (x)/*tan(x) - cot (x)*tan (x)*\-2 - 2*cot (x)/
-----------------------------------------------------------------
3 2
2*cot (x)*tan (x)
$$\frac{\left(2 \tan^{2}{\left(x \right)} + 2\right) \tan{\left(x \right)} \cot^{3}{\left(x \right)} - \left(- 2 \cot^{2}{\left(x \right)} - 2\right) \tan^{2}{\left(x \right)} \cot^{2}{\left(x \right)}}{2 \tan^{2}{\left(x \right)} \cot^{3}{\left(x \right)}}$$
(cot(x)^3*(2 + 2*tan(x)^2)*tan(x) - cot(x)^2*tan(x)^2*(-2 - 2*cot(x)^2))/(2*cot(x)^3*tan(x)^2)
Unión de expresiones racionales
[src]
/ 2 \ / 2 \
\1 + tan (x)/*cot(x) - \-1 - cot (x)/*tan(x)
--------------------------------------------
cot(x)*tan(x)
$$\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \left(- \cot^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} \cot{\left(x \right)}}$$
((1 + tan(x)^2)*cot(x) - (-1 - cot(x)^2)*tan(x))/(cot(x)*tan(x))
Combinatoria
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(1 + cot(x)*tan(x))*(cot(x) + tan(x))
-------------------------------------
cot(x)*tan(x)
$$\frac{\left(\tan{\left(x \right)} \cot{\left(x \right)} + 1\right) \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)}{\tan{\left(x \right)} \cot{\left(x \right)}}$$
(1 + cot(x)*tan(x))*(cot(x) + tan(x))/(cot(x)*tan(x))
Parte trigonométrica
[src]
4 ------------- / pi\ cos|2*x - --| \ 2 /
$$\frac{4}{\cos{\left(2 x - \frac{\pi}{2} \right)}}$$
/ / 2 \ / 2 \\
| 3 | 2*sin (x)| 5 | 2*cos (x)||
|sin (x)*|2 + ---------| sin (x)*|-2 - ---------||
| | 2 | | 2 ||
4 | \ cos (x) / \ sin (x) /|
cos (x)*|----------------------- - ------------------------|
| 3 5 |
\ cos (x) cos (x) /
------------------------------------------------------------
4
2*sin (x)
$$\frac{\left(- \frac{\left(-2 - \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \sin^{5}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + \frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \sin^{3}{\left(x \right)}}{\cos^{3}{\left(x \right)}}\right) \cos^{4}{\left(x \right)}}{2 \sin^{4}{\left(x \right)}}$$
4 -------- sin(2*x)
$$\frac{4}{\sin{\left(2 x \right)}}$$
/ / 2/pi \\ / 2 \\
| | 2*csc |-- - x|| 5/pi \ | 2*csc (x) ||
| 3/pi \ | \2 /| csc |-- - x|*|-2 - ------------||
|csc |-- - x|*|2 + --------------| \2 / | 2/pi \||
| \2 / | 2 | | csc |-- - x|||
4 | \ csc (x) / \ \2 //|
csc (x)*|--------------------------------- - --------------------------------|
| 3 5 |
\ csc (x) csc (x) /
------------------------------------------------------------------------------
4/pi \
2*csc |-- - x|
\2 /
$$\frac{\left(\frac{\left(2 + \frac{2 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(x \right)}} - \frac{\left(- \frac{2 \csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 2\right) \csc^{5}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{5}{\left(x \right)}}\right) \csc^{4}{\left(x \right)}}{2 \csc^{4}{\left(- x + \frac{\pi}{2} \right)}}$$
/ / 2 \ / 2 \\
| 3 | 2*sec (x)| 5 | 2*csc (x)||
|sec (x)*|2 + ---------| sec (x)*|-2 - ---------||
| | 2 | | 2 ||
4 | \ csc (x) / \ sec (x) /|
csc (x)*|----------------------- - ------------------------|
| 3 5 |
\ csc (x) csc (x) /
------------------------------------------------------------
4
2*sec (x)
$$\frac{\left(\frac{\left(2 + \frac{2 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \sec^{3}{\left(x \right)}}{\csc^{3}{\left(x \right)}} - \frac{\left(- \frac{2 \csc^{2}{\left(x \right)}}{\sec^{2}{\left(x \right)}} - 2\right) \sec^{5}{\left(x \right)}}{\csc^{5}{\left(x \right)}}\right) \csc^{4}{\left(x \right)}}{2 \sec^{4}{\left(x \right)}}$$
/ pi\
4*sec|2*x - --|
\ 2 /
$$4 \sec{\left(2 x - \frac{\pi}{2} \right)}$$
/ 2 \
2*\1 + cot (x)/
---------------
cot(x)
$$\frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}}$$
/ 2 \
2*\1 + tan (x)/
---------------
tan(x)
$$\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)}}$$
/ 2 \
|2 + ------- |
| 2 2 |
4 | cot (x) -2 - 2*cot (x)|
cot (x)*|----------- - --------------|
| 3 5 |
\ cot (x) cot (x) /
--------------------------------------
2
$$\frac{\left(\frac{2 + \frac{2}{\cot^{2}{\left(x \right)}}}{\cot^{3}{\left(x \right)}} - \frac{- 2 \cot^{2}{\left(x \right)} - 2}{\cot^{5}{\left(x \right)}}\right) \cot^{4}{\left(x \right)}}{2}$$
3 / 2 \ 5 / 2 \
tan (x)*\2 + 2*tan (x)/ - tan (x)*|-2 - -------|
| 2 |
\ tan (x)/
------------------------------------------------
4
2*tan (x)
$$\frac{- \left(-2 - \frac{2}{\tan^{2}{\left(x \right)}}\right) \tan^{5}{\left(x \right)} + \left(2 \tan^{2}{\left(x \right)} + 2\right) \tan^{3}{\left(x \right)}}{2 \tan^{4}{\left(x \right)}}$$
/ / 2/ pi\\ / 2 \\
| | 2*cos |x - --|| 5/ pi\ | 2*cos (x) ||
| 3/ pi\ | \ 2 /| cos |x - --|*|-2 - ------------||
|cos |x - --|*|2 + --------------| \ 2 / | 2/ pi\||
| \ 2 / | 2 | | cos |x - --|||
4 | \ cos (x) / \ \ 2 //|
cos (x)*|--------------------------------- - --------------------------------|
| 3 5 |
\ cos (x) cos (x) /
------------------------------------------------------------------------------
4/ pi\
2*cos |x - --|
\ 2 /
$$\frac{\left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos^{3}{\left(x \right)}} - \frac{\left(- \frac{2 \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 2\right) \cos^{5}{\left(x - \frac{\pi}{2} \right)}}{\cos^{5}{\left(x \right)}}\right) \cos^{4}{\left(x \right)}}{2 \cos^{4}{\left(x - \frac{\pi}{2} \right)}}$$
4*csc(2*x)
$$4 \csc{\left(2 x \right)}$$
/ / 2 \ / 4 \\
| 10 | sin (2*x)| 6 | 8*sin (x)||
| 32*sin (x)*|-2 - ---------| 8*sin (x)*|2 + ---------||
| | 4 | | 2 ||
4 | \ 2*sin (x)/ \ sin (2*x)/|
sin (2*x)*|- ---------------------------- + -------------------------|
| 5 3 |
\ sin (2*x) sin (2*x) /
----------------------------------------------------------------------
8
32*sin (x)
$$\frac{\left(- \frac{32 \left(-2 - \frac{\sin^{2}{\left(2 x \right)}}{2 \sin^{4}{\left(x \right)}}\right) \sin^{10}{\left(x \right)}}{\sin^{5}{\left(2 x \right)}} + \frac{8 \left(\frac{8 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2\right) \sin^{6}{\left(x \right)}}{\sin^{3}{\left(2 x \right)}}\right) \sin^{4}{\left(2 x \right)}}{32 \sin^{8}{\left(x \right)}}$$
/ / 2 \ / 2/ pi\\\
| 3 | 2*sec (x) | | 2*sec |x - --|||
|sec (x)*|2 + ------------| 5 | \ 2 /||
| | 2/ pi\| sec (x)*|-2 - --------------||
| | sec |x - --|| | 2 ||
4/ pi\ | \ \ 2 // \ sec (x) /|
sec |x - --|*|-------------------------- - -----------------------------|
\ 2 / | 3/ pi\ 5/ pi\ |
| sec |x - --| sec |x - --| |
\ \ 2 / \ 2 / /
-------------------------------------------------------------------------
4
2*sec (x)
$$\frac{\left(- \frac{\left(-2 - \frac{2 \sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec^{5}{\left(x \right)}}{\sec^{5}{\left(x - \frac{\pi}{2} \right)}} + \frac{\left(\frac{2 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2\right) \sec^{3}{\left(x \right)}}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}}\right) \sec^{4}{\left(x - \frac{\pi}{2} \right)}}{2 \sec^{4}{\left(x \right)}}$$
sec(x - pi/2)^4*(sec(x)^3*(2 + 2*sec(x)^2/sec(x - pi/2)^2)/sec(x - pi/2)^3 - sec(x)^5*(-2 - 2*sec(x - pi/2)^2/sec(x)^2)/sec(x - pi/2)^5)/(2*sec(x)^4)