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