Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cot(x)*(-2)-2*cot(x)^3/3-2*cos(x)/(3*sin(x))+2*cos(x)/((3*sin(x)^3))
¿Cómo vas a descomponer esta cot(x)*(-2)-2*cot(x)^3/3-2*cos(x)/(3*sin(x))+2*cos(x)/((3*sin(x)^3)) expresión en fracciones?
Solución
Ha introducido
[src]
3
2*cot (x) 2*cos(x) 2*cos(x)
cot(x)*(-2) - --------- - -------- + ---------
3 3*sin(x) 3
3*sin (x)
$$\left(\left(\left(-2\right) \cot{\left(x \right)} - \frac{2 \cot^{3}{\left(x \right)}}{3}\right) - \frac{2 \cos{\left(x \right)}}{3 \sin{\left(x \right)}}\right) + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$$
cot(x)*(-2) - 2*cot(x)^3/3 - 2*cos(x)/(3*sin(x)) + (2*cos(x))/((3*sin(x)^3))
Respuesta numérica
[src]
-2.0*cot(x) - 0.666666666666667*cot(x)^3 + 0.666666666666667*cos(x)/sin(x)^3 - 0.666666666666667*cos(x)/sin(x)
-2.0*cot(x) - 0.666666666666667*cot(x)^3 + 0.666666666666667*cos(x)/sin(x)^3 - 0.666666666666667*cos(x)/sin(x)
Denominador racional
[src]
3 / / 3 \ \
3*sin (x)*\-6*cos(x) + 3*\-6*cot(x) - 2*cot (x)/*sin(x)/ + 18*cos(x)*sin(x)
---------------------------------------------------------------------------
4
27*sin (x)
$$\frac{3 \left(3 \left(- 2 \cot^{3}{\left(x \right)} - 6 \cot{\left(x \right)}\right) \sin{\left(x \right)} - 6 \cos{\left(x \right)}\right) \sin^{3}{\left(x \right)} + 18 \sin{\left(x \right)} \cos{\left(x \right)}}{27 \sin^{4}{\left(x \right)}}$$
(3*sin(x)^3*(-6*cos(x) + 3*(-6*cot(x) - 2*cot(x)^3)*sin(x)) + 18*cos(x)*sin(x))/(27*sin(x)^4)
Potencias
[src]
3 / I*x -I*x\ / I*x -I*x\
2*cot (x) 8*I*\e + e / 2*I*\e + e /
-2*cot(x) - --------- - ------------------- - ------------------
3 3 / -I*x I*x\
/ -I*x I*x\ 3*\- e + e /
3*\- e + e /
$$- \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} - \frac{2 i \left(e^{i x} + e^{- i x}\right)}{3 \left(e^{i x} - e^{- i x}\right)} - \frac{8 i \left(e^{i x} + e^{- i x}\right)}{3 \left(e^{i x} - e^{- i x}\right)^{3}}$$
3
2*cot (x) 2*cos(x) 2*cos(x)
-2*cot(x) - --------- - -------- + ---------
3 3*sin(x) 3
3*sin (x)
$$- \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{3 \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$$
-2*cot(x) - 2*cot(x)^3/3 - 2*cos(x)/(3*sin(x)) + 2*cos(x)/(3*sin(x)^3)
Unión de expresiones racionales
[src]
/ 2 / / 2 \ \ \
2*\sin (x)*\-cos(x) + \-3 - cot (x)/*cot(x)*sin(x)/ + cos(x)/
-------------------------------------------------------------
3
3*sin (x)
$$\frac{2 \left(\left(\left(- \cot^{2}{\left(x \right)} - 3\right) \sin{\left(x \right)} \cot{\left(x \right)} - \cos{\left(x \right)}\right) \sin^{2}{\left(x \right)} + \cos{\left(x \right)}\right)}{3 \sin^{3}{\left(x \right)}}$$
2*(sin(x)^2*(-cos(x) + (-3 - cot(x)^2)*cot(x)*sin(x)) + cos(x))/(3*sin(x)^3)
Denominador común
[src]
3 2
2*cot (x) -2*cos(x) + 2*sin (x)*cos(x)
-2*cot(x) - --------- - ----------------------------
3 3
3*sin (x)
$$- \frac{2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)}$$
-2*cot(x) - 2*cot(x)^3/3 - (-2*cos(x) + 2*sin(x)^2*cos(x))/(3*sin(x)^3)
Compilar la expresión
[src]
3
2*cot (x) 2*cos(x) 2*cos(x)
-2*cot(x) - --------- - -------- + ---------
3 3*sin(x) 3
3*sin (x)
$$- \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{3 \sin{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$$
-2*cot(x) - 2*cot(x)^3/3 - 2*cos(x)/(3*sin(x)) + 2*cos(x)/(3*sin(x)^3)
Parte trigonométrica
[src]
/x\ / 2/x\\ /x\
tan|-| |1 - tan |-||*tan|-|
2 2 1 \2/ \ \2// \2/
- ------ - --------- - -------- + ------ + --------------------
tan(x) 3 /x\ 3 4/x\ 4/x\
3*tan (x) 3*tan|-| 192*cos |-|*sin |-|
\2/ \4/ \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{192 \sin^{4}{\left(\frac{x}{4} \right)} \cos^{4}{\left(\frac{x}{4} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
/x\
tan|-|
2 2 1 \2/ 8*cos(x)
- ------ - --------- - -------- + ------ - ------------------------
tan(x) 3 /x\ 3 3*(-3*sin(x) + sin(3*x))
3*tan (x) 3*tan|-|
\2/
$$\frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}} - \frac{8 \cos{\left(x \right)}}{3 \left(- 3 \sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}$$
/x\ / 2/x\\ /x\
tan|-| |1 - tan |-||*sec|-|
2 2 1 \2/ \ \2// \2/
- ------ - --------- - -------- + ------ + --------------------
tan(x) 3 /x\ 3 3/x\
3*tan (x) 3*tan|-| 12*sin |-|
\2/ \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \sec{\left(\frac{x}{2} \right)}}{12 \sin^{3}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
/ 2/x\ \
| sec |-| |
3/x pi\ | \2/ | /x\
sec |- - --|*|1 - ------------|*sec|-|
/ pi\ 3/ pi\ /x pi\ /x\ \2 2 / | 2/x pi\| \2/
2*sec|x - --| 2*sec |x - --| sec|- - --| sec|-| | sec |- - --||
\ 2 / \ 2 / \2 2 / \2/ \ \2 2 //
- ------------- - -------------- - ----------- + ------------- + --------------------------------------
sec(x) 3 /x\ /x pi\ 12
3*sec (x) 3*sec|-| 3*sec|- - --|
\2/ \2 2 /
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)} \sec^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{12} + \frac{\sec{\left(\frac{x}{2} \right)}}{3 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} - \frac{2 \sec^{3}{\left(x - \frac{\pi}{2} \right)}}{3 \sec^{3}{\left(x \right)}} - \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{3 \sec{\left(\frac{x}{2} \right)}}$$
/x\
tan|-|
2 2 1 \2/ 8*cos(x)
- ------ - --------- - -------- + ------ + ------------------------
tan(x) 3 /x\ 3 3*(-sin(3*x) + 3*sin(x))
3*tan (x) 3*tan|-|
\2/
$$\frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}} + \frac{8 \cos{\left(x \right)}}{3 \left(3 \sin{\left(x \right)} - \sin{\left(3 x \right)}\right)}$$
3
8*cos(x) 2*cos (x) 2*cos(x)
- -------- - --------- + ---------
3*sin(x) 3 3
3*sin (x) 3*sin (x)
$$- \frac{8 \cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{2 \cos^{3}{\left(x \right)}}{3 \sin^{3}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$$
2
2/x\ / 2/x\\ / 2/x\\
1 - tan |-| |1 + tan |-|| *|1 - tan |-||
2 2 \2/ \ \2// \ \2//
- ------ - --------- - ----------- + ----------------------------
tan(x) 3 /x\ 3/x\
3*tan (x) 3*tan|-| 12*tan |-|
\2/ \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{12 \tan^{3}{\left(\frac{x}{2} \right)}} - \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan{\left(\frac{x}{2} \right)}} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}}$$
4
/ 2/x\\ 3/x\ / 1 \
|1 + tan |-|| *tan |-|*|-1 + -------|
/x\ \ \4// \2/ | 2/x\|
tan|-| | tan |-||
2 2 1 \2/ \ \2//
- ------ - --------- - -------- + ------ + -------------------------------------
tan(x) 3 /x\ 3 4/x\
3*tan (x) 3*tan|-| 192*tan |-|
\2/ \4/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \tan^{3}{\left(\frac{x}{2} \right)}}{192 \tan^{4}{\left(\frac{x}{4} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
2/x\
1 - tan |-|
8 2 \2/
- -------- - --------- + ------------------
3*tan(x) 3 4/x\ 3/x\
3*tan (x) 12*cos |-|*tan |-|
\2/ \2/
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{12 \cos^{4}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}} - \frac{8}{3 \tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}}$$
4
/ 2/x\\ 3/x\ / 1 \
|1 + cot |-|| *cot |-|*|1 - -------|
/x\ \ \4// \2/ | 2/x\|
3 cot|-| | cot |-||
2*cot (x) \2/ 1 \ \2//
-2*cot(x) - --------- - ------ + -------- + ------------------------------------
3 3 /x\ 4
3*cot|-| / 2/x\\
\2/ 12*|-1 + cot |-||
\ \4//
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \cot^{3}{\left(\frac{x}{2} \right)}}{12 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4}} - \frac{\cot{\left(\frac{x}{2} \right)}}{3} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} + \frac{1}{3 \cot{\left(\frac{x}{2} \right)}}$$
3
8*cos(x) 2*cos (x) 2*cos(x)
- ------------- - -------------- + --------------
/ pi\ 3/ pi\ 3/ pi\
3*cos|x - --| 3*cos |x - --| 3*cos |x - --|
\ 2 / \ 2 / \ 2 /
$$- \frac{2 \cos^{3}{\left(x \right)}}{3 \cos^{3}{\left(x - \frac{\pi}{2} \right)}} - \frac{8 \cos{\left(x \right)}}{3 \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{2 \cos{\left(x \right)}}{3 \cos^{3}{\left(x - \frac{\pi}{2} \right)}}$$
/x\ 2/x\
tan|-| 1 - tan |-|
2 2 1 \2/ \2/
- ------ - --------- - -------- + ------ + ---------------------------------
tan(x) 3 /x\ 3 4
3*tan (x) 3*tan|-| / 2/x\\ 8/x\ 3/x\
\2/ 12*|1 - tan |-|| *cos |-|*tan |-|
\ \4// \4/ \2/
$$\frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{12 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \cos^{8}{\left(\frac{x}{4} \right)} \tan^{3}{\left(\frac{x}{2} \right)}}$$
8 2 2*cos(x)
- -------- - --------- + ---------
3*tan(x) 3 3
3*tan (x) 3*sin (x)
$$- \frac{8}{3 \tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}}$$
/x\ /x\ / 2/x\\ /x\
3 cot|-| tan|-| |1 - tan |-||*tan|-|
2*cot (x) \2/ \2/ \ \2// \2/
-2*cot(x) - --------- - ------ + ------ + --------------------
3 3 3 4/x\ 4/x\
192*cos |-|*sin |-|
\4/ \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{192 \sin^{4}{\left(\frac{x}{4} \right)} \cos^{4}{\left(\frac{x}{4} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{\cot{\left(\frac{x}{2} \right)}}{3} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)}$$
2
2/x\ / 2/x\\ / 2/x\\
3 -1 + cot |-| |1 + cot |-|| *|-1 + cot |-||
2*cot (x) \2/ \ \2// \ \2//
-2*cot(x) - --------- - ------------ + -----------------------------
3 /x\ 3/x\
3*cot|-| 12*cot |-|
\2/ \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{12 \cot^{3}{\left(\frac{x}{2} \right)}} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{3 \cot{\left(\frac{x}{2} \right)}} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)}$$
4
/x\ / 2/x\\ / 2/x\\
tan|-| |1 + tan |-|| *|1 - tan |-||
2 2 1 \2/ \ \4// \ \2//
- ------ - --------- - -------- + ------ + ----------------------------
tan(x) 3 /x\ 3 4
3*tan (x) 3*tan|-| / 2/x\\ 3/x\
\2/ 12*|1 - tan |-|| *tan |-|
\ \4// \2/
$$\frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}} + \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4}}{12 \left(1 - \tan^{2}{\left(\frac{x}{4} \right)}\right)^{4} \tan^{3}{\left(\frac{x}{2} \right)}}$$
/ 1 \ /x\
|-1 + -------|*csc|-|
/x\ | 2/x\| \2/
tan|-| | tan |-||
2 2 1 \2/ \ \2//
- ------ - --------- - -------- + ------ + ---------------------
tan(x) 3 /x\ 3 3/x\
3*tan (x) 3*tan|-| 12*cos |-|
\2/ \2/
$$\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}{12 \cos^{3}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
2/x\ 2/x\
2*sin |-| 2*sin |-|
2 2 \2/ sin(x) 1 \2/
- ------ - --------- - --------- - -------------- + ---------------- + ---------
tan(x) 3 3 3*(1 - cos(x)) 2/x\ 3*sin(x)
3*tan (x) 3*sin (x) 6*sin(x)*sin |-|
\2/
$$\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin{\left(x \right)}} - \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin^{3}{\left(x \right)}} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} + \frac{1}{6 \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}} - \frac{\sin{\left(x \right)}}{3 \left(1 - \cos{\left(x \right)}\right)}$$
2/x\
2*sin |-|
2 2 2 2 sin(x) \2/
- ------ - ----------------------- - --------- + ------------------------ - -------------- + ---------
tan(x) 3*(2*sin(x) + sin(2*x)) 3 3*(-sin(2*x) + 2*sin(x)) 3*(1 - cos(x)) 3*sin(x)
3*tan (x)
$$\frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin{\left(x \right)}} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{2}{3 \left(2 \sin{\left(x \right)} + \sin{\left(2 x \right)}\right)} + \frac{2}{3 \left(2 \sin{\left(x \right)} - \sin{\left(2 x \right)}\right)} - \frac{\sin{\left(x \right)}}{3 \left(1 - \cos{\left(x \right)}\right)}$$
3
sin(2*x) 2*cos(x) sin (2*x) 2*cos(x)
- -------- - -------- - ---------- + ---------
2 3*sin(x) 6 3
sin (x) 12*sin (x) 3*sin (x)
$$- \frac{2 \cos{\left(x \right)}}{3 \sin{\left(x \right)}} - \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos{\left(x \right)}}{3 \sin^{3}{\left(x \right)}} - \frac{\sin^{3}{\left(2 x \right)}}{12 \sin^{6}{\left(x \right)}}$$
/ pi\
3 2*sin|x + --|
4*sin(2*x) sin (2*x) \ 2 /
- ---------- - ---------- + -------------
2 6 3
3*sin (x) 12*sin (x) 3*sin (x)
$$- \frac{4 \sin{\left(2 x \right)}}{3 \sin^{2}{\left(x \right)}} + \frac{2 \sin{\left(x + \frac{\pi}{2} \right)}}{3 \sin^{3}{\left(x \right)}} - \frac{\sin^{3}{\left(2 x \right)}}{12 \sin^{6}{\left(x \right)}}$$
2
/ 2/x\\ / 2/x\\
3 |1 + cot |-|| *|-1 + cot |-||
8*cot(x) 2*cot (x) \ \2// \ \2//
- -------- - --------- + -----------------------------
3 3 3/x\
12*cot |-|
\2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{12 \cot^{3}{\left(\frac{x}{2} \right)}} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \frac{8 \cot{\left(x \right)}}{3}$$
/x\ /x\ 3/x\ / 2/x\\
3 cot|-| tan|-| cot |-|*|1 - tan |-||
2*cot (x) \2/ \2/ \2/ \ \2//
-2*cot(x) - --------- - ------ + ------ + --------------------------
3 3 3 4
/ 2/x\\ 8/x\
12*|-1 + cot |-|| *sin |-|
\ \4// \4/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \cot^{3}{\left(\frac{x}{2} \right)}}{12 \left(\cot^{2}{\left(\frac{x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{x}{4} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{\cot{\left(\frac{x}{2} \right)}}{3} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)}$$
/ pi\ / pi\
2*sin|x + --| 3 2*sin|x + --|
sin(2*x) \ 2 / sin (2*x) \ 2 /
- -------- - ------------- - ---------- + -------------
2 3*sin(x) 6 3
sin (x) 12*sin (x) 3*sin (x)
$$- \frac{2 \sin{\left(x + \frac{\pi}{2} \right)}}{3 \sin{\left(x \right)}} - \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \sin{\left(x + \frac{\pi}{2} \right)}}{3 \sin^{3}{\left(x \right)}} - \frac{\sin^{3}{\left(2 x \right)}}{12 \sin^{6}{\left(x \right)}}$$
/ 2/x\ \
| csc |-| |
3/pi x\ | \2/ | /x\
csc |-- - -|*|-1 + ------------|*csc|-|
/x\ /pi x\ \2 2/ | 2/pi x\| \2/
3 csc|-| csc|-- - -| | csc |-- - -||
2*csc(x) 2*csc (x) \2/ \2 2/ \ \2 2//
- ----------- - -------------- - ------------- + ----------- + ---------------------------------------
/pi \ 3/pi \ /pi x\ /x\ 12
csc|-- - x| 3*csc |-- - x| 3*csc|-- - -| 3*csc|-|
\2 / \2 / \2 2/ \2/
$$\frac{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc{\left(\frac{x}{2} \right)} \csc^{3}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{12} - \frac{\csc{\left(\frac{x}{2} \right)}}{3 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{2 \csc^{3}{\left(x \right)}}{3 \csc^{3}{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 \csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{3 \csc{\left(\frac{x}{2} \right)}}$$
2/x pi\
cos |- - --|
\2 2 /
1 - ------------
/x\ /x pi\ 2/x\
3 cos|-| cos|- - --| cos |-|
2*cos(x) 2*cos (x) \2/ \2 2 / \2/
- ----------- - -------------- - ------------- + ----------- + ----------------------
/ pi\ 3/ pi\ /x pi\ /x\ /x\ 3/x pi\
cos|x - --| 3*cos |x - --| 3*cos|- - --| 3*cos|-| 12*cos|-|*cos |- - --|
\ 2 / \ 2 / \2 2 / \2/ \2/ \2 2 /
$$\frac{1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}}{12 \cos{\left(\frac{x}{2} \right)} \cos^{3}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{3 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{2 \cos^{3}{\left(x \right)}}{3 \cos^{3}{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{3 \cos{\left(\frac{x}{2} \right)}}$$
/x\ 2/x\
tan|-| 1 - tan |-|
2 2 1 \2/ \2/
- ------ - --------- - -------- + ------ + ------------------
tan(x) 3 /x\ 3 4/x\ 3/x\
3*tan (x) 3*tan|-| 12*cos |-|*tan |-|
\2/ \2/ \2/
$$\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{12 \cos^{4}{\left(\frac{x}{2} \right)} \tan^{3}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2}{\tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
/ 2/x pi\\
| sec |- - --||
3/x\ | \2 2 /| /x pi\
sec |-|*|-1 + ------------|*sec|- - --|
/ pi\ 3/ pi\ /x pi\ /x\ \2/ | 2/x\ | \2 2 /
2*sec|x - --| 2*sec |x - --| sec|- - --| sec|-| | sec |-| |
\ 2 / \ 2 / \2 2 / \2/ \ \2/ /
- ------------- - -------------- - ----------- + ------------- + ---------------------------------------
sec(x) 3 /x\ /x pi\ 12
3*sec (x) 3*sec|-| 3*sec|- - --|
\2/ \2 2 /
$$\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec^{3}{\left(\frac{x}{2} \right)} \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{12} + \frac{\sec{\left(\frac{x}{2} \right)}}{3 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}} - \frac{2 \sec^{3}{\left(x - \frac{\pi}{2} \right)}}{3 \sec^{3}{\left(x \right)}} - \frac{\sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{3 \sec{\left(\frac{x}{2} \right)}}$$
3 3
8*csc(x) 2*csc (x) 2*csc (x)
- ------------- - -------------- + -------------
/pi \ 3/pi \ /pi \
3*csc|-- - x| 3*csc |-- - x| 3*csc|-- - x|
\2 / \2 / \2 /
$$\frac{2 \csc^{3}{\left(x \right)}}{3 \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 \csc^{3}{\left(x \right)}}{3 \csc^{3}{\left(- x + \frac{\pi}{2} \right)}} - \frac{8 \csc{\left(x \right)}}{3 \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 2 \
2/x\ | sin (x) |
2*sin |-|*|-1 + ---------|
2/x\ \2/ | 4/x\|
3 2*sin |-| | 4*sin |-||
sin(2*x) sin(x) sin (2*x) \2/ \ \2//
- -------- - --------- - ---------- + --------- + --------------------------
2 2/x\ 6 3*sin(x) 3
sin (x) 6*sin |-| 12*sin (x) 3*sin (x)
\2/
$$\frac{2 \left(-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin^{3}{\left(x \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin{\left(x \right)}} - \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\sin^{3}{\left(2 x \right)}}{12 \sin^{6}{\left(x \right)}} - \frac{\sin{\left(x \right)}}{6 \sin^{2}{\left(\frac{x}{2} \right)}}$$
4
/x\ / 2/x\\ / 2/x\\
3 cot|-| |1 + cot |-|| *|-1 + cot |-||
2*cot (x) \2/ 1 \ \4// \ \2//
-2*cot(x) - --------- - ------ + -------- + -----------------------------
3 3 /x\ 3/x\ 4/x\
3*cot|-| 192*cot |-|*cot |-|
\2/ \2/ \4/
$$\frac{\left(\cot^{2}{\left(\frac{x}{4} \right)} + 1\right)^{4} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{192 \cot^{4}{\left(\frac{x}{4} \right)} \cot^{3}{\left(\frac{x}{2} \right)}} - \frac{\cot{\left(\frac{x}{2} \right)}}{3} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} + \frac{1}{3 \cot{\left(\frac{x}{2} \right)}}$$
/ 2/pi x\\
| csc |-- - -||
3/x\ | \2 2/| /pi x\
csc |-|*|1 - ------------|*csc|-- - -|
/x\ /pi x\ \2/ | 2/x\ | \2 2/
3 csc|-| csc|-- - -| | csc |-| |
2*csc(x) 2*csc (x) \2/ \2 2/ \ \2/ /
- ----------- - -------------- - ------------- + ----------- + --------------------------------------
/pi \ 3/pi \ /pi x\ /x\ 12
csc|-- - x| 3*csc |-- - x| 3*csc|-- - -| 3*csc|-|
\2 / \2 / \2 2/ \2/
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc^{3}{\left(\frac{x}{2} \right)} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{12} - \frac{\csc{\left(\frac{x}{2} \right)}}{3 \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - \frac{2 \csc^{3}{\left(x \right)}}{3 \csc^{3}{\left(- x + \frac{\pi}{2} \right)}} - \frac{2 \csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{3 \csc{\left(\frac{x}{2} \right)}}$$
3 3
8*csc(x) 2*csc (x) 2*csc (x)
- -------- - --------- + ---------
3*sec(x) 3 3*sec(x)
3*sec (x)
$$\frac{2 \csc^{3}{\left(x \right)}}{3 \sec{\left(x \right)}} - \frac{2 \csc^{3}{\left(x \right)}}{3 \sec^{3}{\left(x \right)}} - \frac{8 \csc{\left(x \right)}}{3 \sec{\left(x \right)}}$$
3/x\ / 2/x\\
3 tan |-|*|-1 + cot |-||
8*cot(x) 2*cot (x) \2/ \ \2//
- -------- - --------- + ----------------------
3 3 4/x\
12*sin |-|
\2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \tan^{3}{\left(\frac{x}{2} \right)}}{12 \sin^{4}{\left(\frac{x}{2} \right)}} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \frac{8 \cot{\left(x \right)}}{3}$$
/x\ 3/x\ / 2/x\\
3 tan|-| tan |-|*|-1 + cot |-||
2*cot (x) 1 \2/ \2/ \ \2//
-2*cot(x) - --------- - -------- + ------ + ----------------------
3 /x\ 3 4/x\
3*tan|-| 12*sin |-|
\2/ \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \tan^{3}{\left(\frac{x}{2} \right)}}{12 \sin^{4}{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{3} - \frac{2 \cot^{3}{\left(x \right)}}{3} - 2 \cot{\left(x \right)} - \frac{1}{3 \tan{\left(\frac{x}{2} \right)}}$$
/ 4/x\\
| 4*sin |-||
3 | \2/|
2/x\ sin (x)*|1 - ---------|
3 2*sin |-| | 2 |
sin(2*x) sin(x) sin (2*x) \2/ \ sin (x) /
- -------- - --------- - ---------- + --------- + -----------------------
2 2/x\ 6 3*sin(x) 6/x\ 4/pi x\
sin (x) 6*sin |-| 12*sin (x) 96*sin |-|*sin |-- + -|
\2/ \2/ \2 2/
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{3}{\left(x \right)}}{96 \sin^{6}{\left(\frac{x}{2} \right)} \sin^{4}{\left(\frac{x}{2} + \frac{\pi}{2} \right)}} + \frac{2 \sin^{2}{\left(\frac{x}{2} \right)}}{3 \sin{\left(x \right)}} - \frac{\sin{\left(2 x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\sin^{3}{\left(2 x \right)}}{12 \sin^{6}{\left(x \right)}} - \frac{\sin{\left(x \right)}}{6 \sin^{2}{\left(\frac{x}{2} \right)}}$$
/ pi\ 3/ pi\ 3/ pi\
8*sec|x - --| 2*sec |x - --| 2*sec |x - --|
\ 2 / \ 2 / \ 2 /
- ------------- - -------------- + --------------
3*sec(x) 3 3*sec(x)
3*sec (x)
$$\frac{2 \sec^{3}{\left(x - \frac{\pi}{2} \right)}}{3 \sec{\left(x \right)}} - \frac{8 \sec{\left(x - \frac{\pi}{2} \right)}}{3 \sec{\left(x \right)}} - \frac{2 \sec^{3}{\left(x - \frac{\pi}{2} \right)}}{3 \sec^{3}{\left(x \right)}}$$
2/x\
cos |-|
\2/
-1 + ------------
/x\ /x pi\ 2/x pi\
3 cos|-| cos|- - --| cos |- - --|
2*cos(x) 2*cos (x) \2/ \2 2 / \2 2 /
- ----------- - -------------- - ------------- + ----------- + ----------------------
/ pi\ 3/ pi\ /x pi\ /x\ 3/x\ /x pi\
cos|x - --| 3*cos |x - --| 3*cos|- - --| 3*cos|-| 12*cos |-|*cos|- - --|
\ 2 / \ 2 / \2 2 / \2/ \2/ \2 2 /
$$\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{12 \cos^{3}{\left(\frac{x}{2} \right)} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{\cos{\left(\frac{x}{2} \right)}}{3 \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - \frac{2 \cos^{3}{\left(x \right)}}{3 \cos^{3}{\left(x - \frac{\pi}{2} \right)}} - \frac{2 \cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{3 \cos{\left(\frac{x}{2} \right)}}$$
2
/ 2/x\\ / 2/x\\
|1 + tan |-|| *|1 - tan |-||
8 2 \ \2// \ \2//
- -------- - --------- + ----------------------------
3*tan(x) 3 3/x\
3*tan (x) 12*tan |-|
\2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{12 \tan^{3}{\left(\frac{x}{2} \right)}} - \frac{8}{3 \tan{\left(x \right)}} - \frac{2}{3 \tan^{3}{\left(x \right)}}$$
-8/(3*tan(x)) - 2/(3*tan(x)^3) + (1 + tan(x/2)^2)^2*(1 - tan(x/2)^2)/(12*tan(x/2)^3)
Combinatoria
[src]
/ 3 3 2 3 \
-2*\-cos(x) + cot (x)*sin (x) + sin (x)*cos(x) + 3*sin (x)*cot(x)/
------------------------------------------------------------------
3
3*sin (x)
$$- \frac{2 \left(\sin^{3}{\left(x \right)} \cot^{3}{\left(x \right)} + 3 \sin^{3}{\left(x \right)} \cot{\left(x \right)} + \sin^{2}{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)}\right)}{3 \sin^{3}{\left(x \right)}}$$
-2*(-cos(x) + cot(x)^3*sin(x)^3 + sin(x)^2*cos(x) + 3*sin(x)^3*cot(x))/(3*sin(x)^3)