Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(6*pi*x)/(12*pi)+sin(8*pi*x)/(16*pi)
¿Cómo vas a descomponer esta sin(6*pi*x)/(12*pi)+sin(8*pi*x)/(16*pi) expresión en fracciones?
Solución
Ha introducido
[src]
sin(6*pi*x) sin(8*pi*x) ----------- + ----------- 12*pi 16*pi
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\sin{\left(8 \pi x \right)}}{16 \pi}$$
sin((6*pi)*x)/((12*pi)) + sin((8*pi)*x)/((16*pi))
Simplificación general
[src]
sin(6*pi*x) sin(8*pi*x)
----------- + -----------
12 16
-------------------------
pi
$$\frac{\frac{\sin{\left(6 \pi x \right)}}{12} + \frac{\sin{\left(8 \pi x \right)}}{16}}{\pi}$$
(sin(6*pi*x)/12 + sin(8*pi*x)/16)/pi
Respuesta numérica
[src]
0.0265258238486492*sin((6*pi)*x) + 0.0198943678864869*sin((8*pi)*x)
0.0265258238486492*sin((6*pi)*x) + 0.0198943678864869*sin((8*pi)*x)
Denominador racional
[src]
12*pi*sin(8*pi*x) + 16*pi*sin(6*pi*x)
-------------------------------------
2
192*pi
$$\frac{16 \pi \sin{\left(6 \pi x \right)} + 12 \pi \sin{\left(8 \pi x \right)}}{192 \pi^{2}}$$
(12*pi*sin(8*pi*x) + 16*pi*sin(6*pi*x))/(192*pi^2)
Combinatoria
[src]
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
48*pi
$$\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi}$$
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Potencias
[src]
/ -6*pi*I*x 6*pi*I*x\ / -8*pi*I*x 8*pi*I*x\
I*\- e + e / I*\- e + e /
- ---------------------------- - ----------------------------
24*pi 32*pi
$$- \frac{i \left(e^{6 i \pi x} - e^{- 6 i \pi x}\right)}{24 \pi} - \frac{i \left(e^{8 i \pi x} - e^{- 8 i \pi x}\right)}{32 \pi}$$
-i*(-exp(-6*pi*i*x) + exp(6*pi*i*x))/(24*pi) - i*(-exp(-8*pi*i*x) + exp(8*pi*i*x))/(32*pi)
Denominador común
[src]
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
48*pi
$$\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi}$$
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Unión de expresiones racionales
[src]
3*sin(8*pi*x) + 4*sin(6*pi*x)
-----------------------------
48*pi
$$\frac{4 \sin{\left(6 \pi x \right)} + 3 \sin{\left(8 \pi x \right)}}{48 \pi}$$
(3*sin(8*pi*x) + 4*sin(6*pi*x))/(48*pi)
Parte trigonométrica
[src]
tan(3*pi*x) tan(4*pi*x)
----------------------- + -----------------------
/ 2 \ / 2 \
6*pi*\1 + tan (3*pi*x)/ 8*pi*\1 + tan (4*pi*x)/
$$\frac{\tan{\left(4 \pi x \right)}}{8 \pi \left(\tan^{2}{\left(4 \pi x \right)} + 1\right)} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)}$$
sin(6*pi*x) 1 ----------- + ---------------------- 12*pi 16*pi*csc(pi - 8*pi*x)
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \csc{\left(- 8 \pi x + \pi \right)}}$$
2 / 2/pi \\
sin (pi*(1/4 + 4*x))*|1 - cot |-- + 4*pi*x||
sin(6*pi*x) \ \4 //
----------- + --------------------------------------------
12*pi 16*pi
$$\frac{\left(1 - \cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)}\right) \sin^{2}{\left(\pi \left(4 x + \frac{1}{4}\right) \right)}}{16 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
sin(6*pi*x) cos(4*pi*x)*sin(4*pi*x) ----------- + ----------------------- 12*pi 8*pi
$$\frac{\sin{\left(4 \pi x \right)} \cos{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
2*sin(6*pi*x) + 3*cos(4*pi*x)*sin(4*pi*x)
-----------------------------------------
24*pi
$$\frac{3 \sin{\left(4 \pi x \right)} \cos{\left(4 \pi x \right)} + 2 \sin{\left(6 \pi x \right)}}{24 \pi}$$
1 1
----------------- + ---------------------------------
12*pi*csc(6*pi*x) /pi \
8*pi*csc(4*pi*x)*csc|-- - 4*pi*x|
\2 /
$$\frac{1}{12 \pi \csc{\left(6 \pi x \right)}} + \frac{1}{8 \pi \csc{\left(4 \pi x \right)} \csc{\left(- 4 \pi x + \frac{\pi}{2} \right)}}$$
1 1
------------------------ + -----------------
/ pi \ 16*pi*csc(8*pi*x)
12*pi*sec|- -- + 6*pi*x|
\ 2 /
$$\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{16 \pi \csc{\left(8 \pi x \right)}}$$
2/pi \
-1 + tan |-- + 4*pi*x|
cot(3*pi*x) \4 /
----------------------- + -----------------------------
/ 2 \ / 2/pi \\
6*pi*\1 + cot (3*pi*x)/ 16*pi*|1 + tan |-- + 4*pi*x||
\ \4 //
$$\frac{\tan^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} - 1}{16 \pi \left(\tan^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} + 1\right)} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)}$$
cot(3*pi*x) cot(4*pi*x)
----------------------- + -----------------------
/ 2 \ / 2 \
6*pi*\1 + cot (3*pi*x)/ 8*pi*\1 + cot (4*pi*x)/
$$\frac{\cot{\left(4 \pi x \right)}}{8 \pi \left(\cot^{2}{\left(4 \pi x \right)} + 1\right)} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)}$$
2
cot(3*pi*x) cot (2*pi*x)*cot(4*pi*x)
----------------------- + ------------------------
/ 2 \ 2
6*pi*\1 + cot (3*pi*x)/ / 2 \
2*pi*\1 + cot (2*pi*x)/
$$\frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)} + \frac{\cot^{2}{\left(2 \pi x \right)} \cot{\left(4 \pi x \right)}}{2 \pi \left(\cot^{2}{\left(2 \pi x \right)} + 1\right)^{2}}$$
1 1
------------------------ + ----------------------
/ pi \ /pi \
12*pi*sec|- -- + 6*pi*x| 16*pi*sec|-- - 8*pi*x|
\ 2 / \2 /
$$\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{16 \pi \sec{\left(- 8 \pi x + \frac{\pi}{2} \right)}}$$
1 1
------------------------ + ------------------------
/ pi \ / pi \
12*pi*sec|- -- + 6*pi*x| 16*pi*sec|- -- + 8*pi*x|
\ 2 / \ 2 /
$$\frac{1}{16 \pi \sec{\left(8 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}}$$
1 1 ----------------- + ---------------------- 12*pi*csc(6*pi*x) 16*pi*csc(pi - 8*pi*x)
$$\frac{1}{16 \pi \csc{\left(- 8 \pi x + \pi \right)}} + \frac{1}{12 \pi \csc{\left(6 \pi x \right)}}$$
2 sin(6*pi*x) cos (4*pi*x)*tan(4*pi*x) ----------- + ------------------------ 12*pi 8*pi
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\cos^{2}{\left(4 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi}$$
sin(6*pi*x) sin(4*pi*x) ----------- + ---------------- 12*pi 8*pi*sec(4*pi*x)
$$\frac{\sin{\left(4 \pi x \right)}}{8 \pi \sec{\left(4 \pi x \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
2 sin(6*pi*x) sin (4*pi*x)*cot(4*pi*x) ----------- + ------------------------ 12*pi 8*pi
$$\frac{\sin^{2}{\left(4 \pi x \right)} \cot{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
/ pi \ / pi \
cos|- -- + 6*pi*x| cos(4*pi*x)*cos|- -- + 4*pi*x|
\ 2 / \ 2 /
------------------ + ------------------------------
12*pi 8*pi
$$\frac{\cos{\left(4 \pi x \right)} \cos{\left(4 \pi x - \frac{\pi}{2} \right)}}{8 \pi} + \frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi}$$
2
/ 2 \
tan(3*pi*x) \1 - tan (2*pi*x)/ *tan(4*pi*x)
----------------------- + -------------------------------
/ 2 \ 2
6*pi*\1 + tan (3*pi*x)/ / 2 \
8*pi*\1 + tan (2*pi*x)/
$$\frac{\left(1 - \tan^{2}{\left(2 \pi x \right)}\right)^{2} \tan{\left(4 \pi x \right)}}{8 \pi \left(\tan^{2}{\left(2 \pi x \right)} + 1\right)^{2}} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)}$$
2
/ 2 \
cot(3*pi*x) \-1 + cot (2*pi*x)/
----------------------- + ------------------------------------
/ 2 \ 2
6*pi*\1 + cot (3*pi*x)/ / 2 \
8*pi*\1 + cot (2*pi*x)/ *cot(4*pi*x)
$$\frac{\left(\cot^{2}{\left(2 \pi x \right)} - 1\right)^{2}}{8 \pi \left(\cot^{2}{\left(2 \pi x \right)} + 1\right)^{2} \cot{\left(4 \pi x \right)}} + \frac{\cot{\left(3 \pi x \right)}}{6 \pi \left(\cot^{2}{\left(3 \pi x \right)} + 1\right)}$$
/ pi \
cos|- -- + 6*pi*x|
\ 2 / sin(8*pi*x)
------------------ + -----------
12*pi 16*pi
$$\frac{\sin{\left(8 \pi x \right)}}{16 \pi} + \frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi}$$
2
/ 2 \ 4
sin(6*pi*x) \1 - tan (2*pi*x)/ *cos (2*pi*x)*tan(4*pi*x)
----------- + --------------------------------------------
12*pi 8*pi
$$\frac{\left(1 - \tan^{2}{\left(2 \pi x \right)}\right)^{2} \cos^{4}{\left(2 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
2
tan(3*pi*x) tan (2*pi*x)
----------------------- + ------------------------------------
/ 2 \ 2
6*pi*\1 + tan (3*pi*x)/ / 2 \
2*pi*\1 + tan (2*pi*x)/ *tan(4*pi*x)
$$\frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)} + \frac{\tan^{2}{\left(2 \pi x \right)}}{2 \pi \left(\tan^{2}{\left(2 \pi x \right)} + 1\right)^{2} \tan{\left(4 \pi x \right)}}$$
/ pi \
cos|- -- + 8*pi*x|
sin(6*pi*x) \ 2 /
----------- + ------------------
12*pi 16*pi
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{\cos{\left(8 \pi x - \frac{\pi}{2} \right)}}{16 \pi}$$
sin(6*pi*x) 1
----------- + ----------------------
12*pi /pi \
16*pi*sec|-- - 8*pi*x|
\2 /
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \sec{\left(- 8 \pi x + \frac{\pi}{2} \right)}}$$
sin(6*pi*x) 1 ----------- + ----------------- 12*pi 16*pi*csc(8*pi*x)
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \csc{\left(8 \pi x \right)}}$$
sin(6*pi*x) 1
----------- + ------------------------
12*pi / pi \
16*pi*sec|- -- + 8*pi*x|
\ 2 /
$$\frac{\sin{\left(6 \pi x \right)}}{12 \pi} + \frac{1}{16 \pi \sec{\left(8 \pi x - \frac{\pi}{2} \right)}}$$
2 2/pi \
sin (4*pi*x)*sin |-- + 4*pi*x|
sin(6*pi*x) \2 /
----------- + ------------------------------
12*pi 4*pi*sin(8*pi*x)
$$\frac{\sin^{2}{\left(4 \pi x \right)} \sin^{2}{\left(4 \pi x + \frac{\pi}{2} \right)}}{4 \pi \sin{\left(8 \pi x \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
1 1
------------------------ + -----------------------------------
/ pi \ / pi \
12*pi*sec|- -- + 6*pi*x| 8*pi*sec(4*pi*x)*sec|- -- + 4*pi*x|
\ 2 / \ 2 /
$$\frac{1}{12 \pi \sec{\left(6 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{8 \pi \sec{\left(4 \pi x \right)} \sec{\left(4 \pi x - \frac{\pi}{2} \right)}}$$
2
/ 2 \ 4
sin(6*pi*x) \-1 + cot (2*pi*x)/ *sin (2*pi*x)*tan(4*pi*x)
----------- + ---------------------------------------------
12*pi 8*pi
$$\frac{\left(\cot^{2}{\left(2 \pi x \right)} - 1\right)^{2} \sin^{4}{\left(2 \pi x \right)} \tan{\left(4 \pi x \right)}}{8 \pi} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
/ pi \ / pi \
cos|- -- + 6*pi*x| cos|- -- + 8*pi*x|
\ 2 / \ 2 /
------------------ + ------------------
12*pi 16*pi
$$\frac{\cos{\left(6 \pi x - \frac{\pi}{2} \right)}}{12 \pi} + \frac{\cos{\left(8 \pi x - \frac{\pi}{2} \right)}}{16 \pi}$$
2/pi \
1 - cot |-- + 4*pi*x|
tan(3*pi*x) \4 /
----------------------- + -----------------------------
/ 2 \ / 2/pi \\
6*pi*\1 + tan (3*pi*x)/ 16*pi*|1 + cot |-- + 4*pi*x||
\ \4 //
$$\frac{1 - \cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)}}{16 \pi \left(\cot^{2}{\left(4 \pi x + \frac{\pi}{4} \right)} + 1\right)} + \frac{\tan{\left(3 \pi x \right)}}{6 \pi \left(\tan^{2}{\left(3 \pi x \right)} + 1\right)}$$
1 1 ----------------- + ----------------- 12*pi*csc(6*pi*x) 16*pi*csc(8*pi*x)
$$\frac{1}{16 \pi \csc{\left(8 \pi x \right)}} + \frac{1}{12 \pi \csc{\left(6 \pi x \right)}}$$
sin(6*pi*x) sin(4*pi*x)
----------- + ---------------------
12*pi /pi \
8*pi*csc|-- - 4*pi*x|
\2 /
$$\frac{\sin{\left(4 \pi x \right)}}{8 \pi \csc{\left(- 4 \pi x + \frac{\pi}{2} \right)}} + \frac{\sin{\left(6 \pi x \right)}}{12 \pi}$$
sin(6*pi*x)/(12*pi) + sin(4*pi*x)/(8*pi*csc(pi/2 - 4*pi*x))