Sr Examen
¿Cómo vas a descomponer esta tan(x)/(tan(2*x)-tg(x)) expresión en fracciones?
Solución
Ha introducido
[src]
tan(x) ----------------- tan(2*x) - tan(x)
$$\frac{\tan{\left(x \right)}}{- \tan{\left(x \right)} + \tan{\left(2 x \right)}}$$
tan(x)/(tan(2*x) - tan(x))
Parte trigonométrica
[src]
1 -------------------------------------------------- / 1 sec(2*x) \ / pi\ |- ----------- + --------------------|*sec|x - --| | / pi\ / pi\| \ 2 / | sec|x - --| sec(x)*sec|2*x - --|| \ \ 2 / \ 2 //
$$\frac{1}{\left(- \frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(2 x \right)}}{\sec{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
1 ---------------------------------------- / /pi \ \ | csc|-- - 2*x| | | 1 \2 / | |- ------ + --------------------|*csc(x) | csc(x) /pi \| | csc(2*x)*csc|-- - x|| \ \2 //
$$\frac{1}{\left(\frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/x\
2*tan|-|
\2/
----------------------------------
/ 2/x\\
|1 - tan |-||*(-tan(x) + tan(2*x))
\ \2//
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(- \tan{\left(x \right)} + \tan{\left(2 x \right)}\right)}$$
sin(x) ------------------------- -sin(x) + cos(x)*tan(2*x)
$$\frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(2 x \right)}}$$
/x\
2*cot|-|
\2/
----------------------------------
/ 2/x\\ / 1 1 \
|-1 + cot |-||*|-------- - ------|
\ \2// \cot(2*x) cot(x)/
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\frac{1}{\cot{\left(2 x \right)}} - \frac{1}{\cot{\left(x \right)}}\right)}$$
sec(x) -------------------------- /sec(2*x) sec(x)\ |-------- - ------|*csc(x) \csc(2*x) csc(x)/
$$\frac{\sec{\left(x \right)}}{\left(\frac{\sec{\left(2 x \right)}}{\csc{\left(2 x \right)}} - \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
/ pi\
cos|x - --|
\ 2 /
------------------------------------
/ / pi\ / pi\\
|cos|2*x - --| cos|x - --||
| \ 2 / \ 2 /|
|------------- - -----------|*cos(x)
\ cos(2*x) cos(x) /
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\left(\frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)}} - \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos{\left(x \right)}}$$
2
2*sin (x)
------------------------------------
/ 2 2 \
| 2*sin (x) 2*sin (2*x)|
|- --------- + -----------|*sin(2*x)
\ sin(2*x) sin(4*x) /
$$\frac{2 \sin^{2}{\left(x \right)}}{\left(- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{2 \sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}}\right) \sin{\left(2 x \right)}}$$
/ pi\
cos|x - --|
\ 2 /
------------------------------------
/ pi\
cos(x)*cos|2*x - --|
/ pi\ \ 2 /
- cos|x - --| + --------------------
\ 2 / cos(2*x)
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\frac{\cos{\left(x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}}{\cos{\left(2 x \right)}} - \cos{\left(x - \frac{\pi}{2} \right)}}$$
/x\
2*tan|-|
\2/
------------------------------------------------------
/ /x\ / 2/x\\ \
| 2*tan|-| |1 - tan |-||*tan(2*x)|
/ 2/x\\ | \2/ \ \2// |
|1 + tan |-||*|- ----------- + ----------------------|
\ \2// | 2/x\ 2/x\ |
| 1 + tan |-| 1 + tan |-| |
\ \2/ \2/ /
$$\frac{2 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(2 x \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
-sin(x) ------------------------- -cos(x)*tan(2*x) + sin(x)
$$- \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)} \tan{\left(2 x \right)}}$$
sin(x) --------------------------------------- / 2 2 \ | 2*sin (x) 2*sin (2*x)| / pi\ |- --------- + -----------|*sin|x + --| \ sin(2*x) sin(4*x) / \ 2 /
$$\frac{\sin{\left(x \right)}}{\left(- \frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{2 \sin^{2}{\left(2 x \right)}}{\sin{\left(4 x \right)}}\right) \sin{\left(x + \frac{\pi}{2} \right)}}$$
sin(x) -------------------------- /sin(2*x) sin(x)\ |-------- - ------|*cos(x) \cos(2*x) cos(x)/
$$\frac{\sin{\left(x \right)}}{\left(- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}\right) \cos{\left(x \right)}}$$
/x\
2*cot|-|
\2/
------------------------------------------------------
/ /x\ 2/x\ \
| 2*cot|-| -1 + cot |-| |
/ 2/x\\ | \2/ \2/ |
|1 + cot |-||*|- ----------- + ----------------------|
\ \2// | 2/x\ / 2/x\\ |
| 1 + cot |-| |1 + cot |-||*cot(2*x)|
\ \2/ \ \2// /
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \cot{\left(2 x \right)}} - \frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
cos(2*x)
$$\cos{\left(2 x \right)}$$
/x\
2*cot|-|
\2/
-----------------------------------
/ 2/x\\
|-1 + cot |-||*(-tan(x) + tan(2*x))
\ \2//
$$\frac{2 \cot{\left(\frac{x}{2} \right)}}{\left(- \tan{\left(x \right)} + \tan{\left(2 x \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}$$
sin(x)
---------------------------------
2 / pi\
2*sin (2*x)*sin|x + --|
\ 2 /
-sin(x) + -----------------------
sin(4*x)
$$\frac{\sin{\left(x \right)}}{- \sin{\left(x \right)} + \frac{2 \sin^{2}{\left(2 x \right)} \sin{\left(x + \frac{\pi}{2} \right)}}{\sin{\left(4 x \right)}}}$$
/pi \
csc|-- - x|
\2 /
------------------------------------
/ /pi \ /pi \\
|csc|-- - 2*x| csc|-- - x||
| \2 / \2 /|
|------------- - -----------|*csc(x)
\ csc(2*x) csc(x) /
$$\frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\left(\frac{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc{\left(2 x \right)}} - \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(x \right)}}$$
sec(x)*sin(x) ------------------ -tan(x) + tan(2*x)
$$\frac{\sin{\left(x \right)} \sec{\left(x \right)}}{- \tan{\left(x \right)} + \tan{\left(2 x \right)}}$$
sec(x) ----------------------------------------- / sec(2*x) sec(x) \ / pi\ |------------- - -----------|*sec|x - --| | / pi\ / pi\| \ 2 / |sec|2*x - --| sec|x - --|| \ \ 2 / \ 2 //
$$\frac{\sec{\left(x \right)}}{\left(- \frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\sec{\left(2 x \right)}}{\sec{\left(2 x - \frac{\pi}{2} \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
1 -------------------------- / 1 1 \ |-------- - ------|*cot(x) \cot(2*x) cot(x)/
$$\frac{1}{\left(\frac{1}{\cot{\left(2 x \right)}} - \frac{1}{\cot{\left(x \right)}}\right) \cot{\left(x \right)}}$$
1/((1/cot(2*x) - 1/cot(x))*cot(x))
Potencias
[src]
/ I*x -I*x\
I*\- e + e /
------------------------------------------------------------
/ / 2*I*x -2*I*x\ / I*x -I*x\\
|I*\- e + e / I*\- e + e /| / I*x -I*x\
|---------------------- - ------------------|*\e + e /
| -2*I*x 2*I*x I*x -I*x |
\ e + e e + e /
$$\frac{i \left(- e^{i x} + e^{- i x}\right)}{\left(- \frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + \frac{i \left(- e^{2 i x} + e^{- 2 i x}\right)}{e^{2 i x} + e^{- 2 i x}}\right) \left(e^{i x} + e^{- i x}\right)}$$
i*(-exp(i*x) + exp(-i*x))/((i*(-exp(2*i*x) + exp(-2*i*x))/(exp(-2*i*x) + exp(2*i*x)) - i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))*(exp(i*x) + exp(-i*x)))