Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(pi*x)/(2*pi)+sin(3*pi*x)/(6*pi)
¿Cómo vas a descomponer esta sin(pi*x)/(2*pi)+sin(3*pi*x)/(6*pi) expresión en fracciones?
Solución
Ha introducido
[src]
sin(pi*x) sin(3*pi*x) --------- + ----------- 2*pi 6*pi
$$\frac{\sin{\left(\pi x \right)}}{2 \pi} + \frac{\sin{\left(3 \pi x \right)}}{6 \pi}$$
sin(pi*x)/((2*pi)) + sin((3*pi)*x)/((6*pi))
Simplificación general
[src]
3*sin(pi*x) + sin(3*pi*x)
-------------------------
6*pi
$$\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi}$$
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Respuesta numérica
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0.0530516476972984*sin((3*pi)*x) + 0.159154943091895*sin(pi*x)
0.0530516476972984*sin((3*pi)*x) + 0.159154943091895*sin(pi*x)
Denominador racional
[src]
2*pi*sin(3*pi*x) + 6*pi*sin(pi*x)
---------------------------------
2
12*pi
$$\frac{6 \pi \sin{\left(\pi x \right)} + 2 \pi \sin{\left(3 \pi x \right)}}{12 \pi^{2}}$$
(2*pi*sin(3*pi*x) + 6*pi*sin(pi*x))/(12*pi^2)
Denominador común
[src]
3*sin(pi*x) + sin(3*pi*x)
-------------------------
6*pi
$$\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi}$$
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Combinatoria
[src]
3*sin(pi*x) + sin(3*pi*x)
-------------------------
6*pi
$$\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi}$$
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Potencias
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/ -pi*I*x pi*I*x\ / -3*pi*I*x 3*pi*I*x\
I*\- e + e / I*\- e + e /
- ------------------------ - ----------------------------
4*pi 12*pi
$$- \frac{i \left(e^{i \pi x} - e^{- i \pi x}\right)}{4 \pi} - \frac{i \left(e^{3 i \pi x} - e^{- 3 i \pi x}\right)}{12 \pi}$$
-i*(-exp(-pi*i*x) + exp(pi*i*x))/(4*pi) - i*(-exp(-3*pi*i*x) + exp(3*pi*i*x))/(12*pi)
Unión de expresiones racionales
[src]
3*sin(pi*x) + sin(3*pi*x)
-------------------------
6*pi
$$\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi}$$
(3*sin(pi*x) + sin(3*pi*x))/(6*pi)
Parte trigonométrica
[src]
3*sin(pi*x) + sin(3*pi*x)
-------------------------
6*pi
$$\frac{3 \sin{\left(\pi x \right)} + \sin{\left(3 \pi x \right)}}{6 \pi}$$
/pi*x\ /3*pi*x\
tan|----| tan|------|
\ 2 / \ 2 /
------------------- + -----------------------
/ 2/pi*x\\ / 2/3*pi*x\\
pi*|1 + tan |----|| 3*pi*|1 + tan |------||
\ \ 2 // \ \ 2 //
$$\frac{\tan{\left(\frac{3 \pi x}{2} \right)}}{3 \pi \left(\tan^{2}{\left(\frac{3 \pi x}{2} \right)} + 1\right)} + \frac{\tan{\left(\frac{\pi x}{2} \right)}}{\pi \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}$$
/3*pi*x\ /pi*x\
2*tan|------| 6*tan|----|
\ 2 / \ 2 /
---------------- + --------------
2/3*pi*x\ 2/pi*x\
1 + tan |------| 1 + tan |----|
\ 2 / \ 2 /
---------------------------------
6*pi
$$\frac{\frac{2 \tan{\left(\frac{3 \pi x}{2} \right)}}{\tan^{2}{\left(\frac{3 \pi x}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{\pi x}{2} \right)}}{\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{6 \pi}$$
/ pi \ / pi \
3*cos|- -- + pi*x| + cos|- -- + 3*pi*x|
\ 2 / \ 2 /
---------------------------------------
6*pi
$$\frac{3 \cos{\left(\pi x - \frac{\pi}{2} \right)} + \cos{\left(3 \pi x - \frac{\pi}{2} \right)}}{6 \pi}$$
/pi*x\ /3*pi*x\
cot|----| cot|------|
\ 2 / \ 2 /
------------------- + -----------------------
/ 2/pi*x\\ / 2/3*pi*x\\
pi*|1 + cot |----|| 3*pi*|1 + cot |------||
\ \ 2 // \ \ 2 //
$$\frac{\cot{\left(\frac{3 \pi x}{2} \right)}}{3 \pi \left(\cot^{2}{\left(\frac{3 \pi x}{2} \right)} + 1\right)} + \frac{\cot{\left(\frac{\pi x}{2} \right)}}{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)}$$
1 3
------------------ + ----------------
/ pi \ / pi \
sec|- -- + 3*pi*x| sec|- -- + pi*x|
\ 2 / \ 2 /
-------------------------------------
6*pi
$$\frac{\frac{1}{\sec{\left(3 \pi x - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(\pi x - \frac{\pi}{2} \right)}}}{6 \pi}$$
1 3
----------- + ---------
csc(3*pi*x) csc(pi*x)
-----------------------
6*pi
$$\frac{\frac{1}{\csc{\left(3 \pi x \right)}} + \frac{3}{\csc{\left(\pi x \right)}}}{6 \pi}$$
1 1
--------------------- + -----------------------
/ pi \ / pi \
2*pi*sec|- -- + pi*x| 6*pi*sec|- -- + 3*pi*x|
\ 2 / \ 2 /
$$\frac{1}{6 \pi \sec{\left(3 \pi x - \frac{\pi}{2} \right)}} + \frac{1}{2 \pi \sec{\left(\pi x - \frac{\pi}{2} \right)}}$$
1 1 -------------- + ---------------- 2*pi*csc(pi*x) 6*pi*csc(3*pi*x)
$$\frac{1}{6 \pi \csc{\left(3 \pi x \right)}} + \frac{1}{2 \pi \csc{\left(\pi x \right)}}$$
/ pi \ / pi \
cos|- -- + pi*x| cos|- -- + 3*pi*x|
\ 2 / \ 2 /
---------------- + ------------------
2*pi 6*pi
$$\frac{\cos{\left(\pi x - \frac{\pi}{2} \right)}}{2 \pi} + \frac{\cos{\left(3 \pi x - \frac{\pi}{2} \right)}}{6 \pi}$$
/3*pi*x\ /pi*x\
2*cot|------| 6*cot|----|
\ 2 / \ 2 /
---------------- + --------------
2/3*pi*x\ 2/pi*x\
1 + cot |------| 1 + cot |----|
\ 2 / \ 2 /
---------------------------------
6*pi
$$\frac{\frac{2 \cot{\left(\frac{3 \pi x}{2} \right)}}{\cot^{2}{\left(\frac{3 \pi x}{2} \right)} + 1} + \frac{6 \cot{\left(\frac{\pi x}{2} \right)}}{\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1}}{6 \pi}$$
(2*cot(3*pi*x/2)/(1 + cot(3*pi*x/2)^2) + 6*cot(pi*x/2)/(1 + cot(pi*x/2)^2))/(6*pi)