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