Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)^2-cos(x)^2+10/(x*(x^2-5*x+1))+10*(5-2*x)*log(x)/(x^2-5*x+1)^2
¿Cómo vas a descomponer esta sin(x)^2-cos(x)^2+10/(x*(x^2-5*x+1))+10*(5-2*x)*log(x)/(x^2-5*x+1)^2 expresión en fracciones?
Solución
Ha introducido
[src]
2 2 10 10*(5 - 2*x)*log(x)
sin (x) - cos (x) + ---------------- + -------------------
/ 2 \ 2
x*\x - 5*x + 1/ / 2 \
\x - 5*x + 1/
$$\frac{10 \left(5 - 2 x\right) \log{\left(x \right)}}{\left(\left(x^{2} - 5 x\right) + 1\right)^{2}} + \left(\left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + \frac{10}{x \left(\left(x^{2} - 5 x\right) + 1\right)}\right)$$
sin(x)^2 - cos(x)^2 + 10/((x*(x^2 - 5*x + 1))) + ((10*(5 - 2*x))*log(x))/(x^2 - 5*x + 1)^2
Simplificación general
[src]
2
2 / 2 \
10 - 50*x + 10*x - x*\1 + x - 5*x/ *cos(2*x) - 10*x*(-5 + 2*x)*log(x)
-----------------------------------------------------------------------
2
/ 2 \
x*\1 + x - 5*x/
$$\frac{10 x^{2} - 10 x \left(2 x - 5\right) \log{\left(x \right)} - x \left(x^{2} - 5 x + 1\right)^{2} \cos{\left(2 x \right)} - 50 x + 10}{x \left(x^{2} - 5 x + 1\right)^{2}}$$
(10 - 50*x + 10*x^2 - x*(1 + x^2 - 5*x)^2*cos(2*x) - 10*x*(-5 + 2*x)*log(x))/(x*(1 + x^2 - 5*x)^2)
Respuesta numérica
[src]
sin(x)^2 - cos(x)^2 + 10.0/(x*(1.0 + x^2 - 5.0*x)) + 0.04*(50.0 - 20.0*x)*log(x)/(0.2 - x + 0.2*x^2)^2
sin(x)^2 - cos(x)^2 + 10.0/(x*(1.0 + x^2 - 5.0*x)) + 0.04*(50.0 - 20.0*x)*log(x)/(0.2 - x + 0.2*x^2)^2
Unión de expresiones racionales
[src]
/ / 2 2 \\
(1 + x*(-5 + x))*\10 + x*(1 + x*(-5 + x))*\sin (x) - cos (x)// + 10*x*(5 - 2*x)*log(x)
--------------------------------------------------------------------------------------
2
x*(1 + x*(-5 + x))
$$\frac{10 x \left(5 - 2 x\right) \log{\left(x \right)} + \left(x \left(x - 5\right) + 1\right) \left(x \left(x \left(x - 5\right) + 1\right) \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) + 10\right)}{x \left(x \left(x - 5\right) + 1\right)^{2}}$$
((1 + x*(-5 + x))*(10 + x*(1 + x*(-5 + x))*(sin(x)^2 - cos(x)^2)) + 10*x*(5 - 2*x)*log(x))/(x*(1 + x*(-5 + x))^2)
Denominador racional
[src]
2
/ 2 \ / / 2 2 \ / 2 \\ / 2 \
\1 + x - 5*x/ *\10 + x*\sin (x) - cos (x)/*\1 + x - 5*x// + x*(50 - 20*x)*\1 + x - 5*x/*log(x)
-------------------------------------------------------------------------------------------------
3
/ 2 \
x*\1 + x - 5*x/
$$\frac{x \left(50 - 20 x\right) \left(x^{2} - 5 x + 1\right) \log{\left(x \right)} + \left(x \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \left(x^{2} - 5 x + 1\right) + 10\right) \left(x^{2} - 5 x + 1\right)^{2}}{x \left(x^{2} - 5 x + 1\right)^{3}}$$
((1 + x^2 - 5*x)^2*(10 + x*(sin(x)^2 - cos(x)^2)*(1 + x^2 - 5*x)) + x*(50 - 20*x)*(1 + x^2 - 5*x)*log(x))/(x*(1 + x^2 - 5*x)^3)
Combinatoria
[src]
/ 2 2 5 2 2 5 2 3 2 2 2 4 2 2 2 4 2 2 3 2 \
-\-10 - 10*x + 50*x + x*cos (x) + x *cos (x) - x*sin (x) - x *sin (x) - 50*x*log(x) - 27*x *sin (x) - 10*x *cos (x) - 10*x *cos (x) + 10*x *sin (x) + 10*x *sin (x) + 20*x *log(x) + 27*x *cos (x)/
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2
/ 2 \
x*\1 + x - 5*x/
$$- \frac{- x^{5} \sin^{2}{\left(x \right)} + x^{5} \cos^{2}{\left(x \right)} + 10 x^{4} \sin^{2}{\left(x \right)} - 10 x^{4} \cos^{2}{\left(x \right)} - 27 x^{3} \sin^{2}{\left(x \right)} + 27 x^{3} \cos^{2}{\left(x \right)} + 20 x^{2} \log{\left(x \right)} + 10 x^{2} \sin^{2}{\left(x \right)} - 10 x^{2} \cos^{2}{\left(x \right)} - 10 x^{2} - 50 x \log{\left(x \right)} - x \sin^{2}{\left(x \right)} + x \cos^{2}{\left(x \right)} + 50 x - 10}{x \left(x^{2} - 5 x + 1\right)^{2}}$$
-(-10 - 10*x^2 + 50*x + x*cos(x)^2 + x^5*cos(x)^2 - x*sin(x)^2 - x^5*sin(x)^2 - 50*x*log(x) - 27*x^3*sin(x)^2 - 10*x^2*cos(x)^2 - 10*x^4*cos(x)^2 + 10*x^2*sin(x)^2 + 10*x^4*sin(x)^2 + 20*x^2*log(x) + 27*x^3*cos(x)^2)/(x*(1 + x^2 - 5*x)^2)
Compilar la expresión
[src]
2 2 10 (50 - 20*x)*log(x)
sin (x) - cos (x) + ---------------- + ------------------
/ 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
sin(x)^2 - cos(x)^2 + 10/(x*(1 + x^2 - 5*x)) + (50 - 20*x)*log(x)/(1 + x^2 - 5*x)^2
Denominador común
[src]
2 2
2 2 -10 - 10*x + 50*x - 50*x*log(x) + 20*x *log(x)
sin (x) - cos (x) - -----------------------------------------------
5 2 4 3
x + x - 10*x - 10*x + 27*x
$$\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} - \frac{20 x^{2} \log{\left(x \right)} - 10 x^{2} - 50 x \log{\left(x \right)} + 50 x - 10}{x^{5} - 10 x^{4} + 27 x^{3} - 10 x^{2} + x}$$
sin(x)^2 - cos(x)^2 - (-10 - 10*x^2 + 50*x - 50*x*log(x) + 20*x^2*log(x))/(x + x^5 - 10*x^2 - 10*x^4 + 27*x^3)
Parte trigonométrica
[src]
/pi \ 10 (50 - 20*x)*log(x)
- sin|-- + 2*x| + ---------------- + ------------------
\2 / / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \sin{\left(2 x + \frac{\pi}{2} \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2
1 - tan (x) 10 (50 - 20*x)*log(x)
- ----------- + ---------------- + ------------------
2 / 2 \ 2
1 + tan (x) x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$- \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + \frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 10 (50 - 20*x)*log(x)
- -------- + ---------------- + ------------------
sec(2*x) / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{1}{\sec{\left(2 x \right)}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2
/ 2/x\\ 2/x\
|-1 + cot |-|| 4*cot |-|
\ \2// \2/ 10 (50 - 20*x)*log(x)
- --------------- + -------------- + ---------------- + ------------------
2 2 / 2 \ 2
/ 2/x\\ / 2/x\\ x*\1 + x - 5*x/ / 2 \
|1 + cot |-|| |1 + cot |-|| \1 + x - 5*x/
\ \2// \ \2//
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
10*(5 - 2*x)*log(x) 10
-cos(2*x) + ------------------- + ----------------
2 / 2 \
/ 2 \ x*\x - 5*x + 1/
\x - 5*x + 1/
$$\frac{10 \left(5 - 2 x\right) \log{\left(x \right)}}{\left(\left(x^{2} - 5 x\right) + 1\right)^{2}} - \cos{\left(2 x \right)} + \frac{10}{x \left(\left(x^{2} - 5 x\right) + 1\right)}$$
2
/ 2/x\\ 2/x\
|1 - tan |-|| 4*tan |-|
\ \2// \2/ 10 (50 - 20*x)*log(x)
- -------------- + -------------- + ---------------- + ------------------
2 2 / 2 \ 2
/ 2/x\\ / 2/x\\ x*\1 + x - 5*x/ / 2 \
|1 + tan |-|| |1 + tan |-|| \1 + x - 5*x/
\ \2// \ \2//
$$- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 1 10 (50 - 20*x)*log(x)
------- - ------------ + ---------------- + ------------------
2 2/pi \ / 2 \ 2
csc (x) csc |-- - x| x*\1 + x - 5*x/ / 2 \
\2 / \1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2 2/ pi\ 10 (50 - 20*x)*log(x)
sin (x) - sin |x + --| + ---------------- + ------------------
\ 2 / / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \sin^{2}{\left(x \right)} - \sin^{2}{\left(x + \frac{\pi}{2} \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 10 (50 - 20*x)*log(x)
- ------------- + ---------------- + ------------------
/pi \ / 2 \ 2
csc|-- - 2*x| x*\1 + x - 5*x/ / 2 \
\2 / \1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2/ pi\ 2 10 (50 - 20*x)*log(x)
cos |x - --| - cos (x) + ---------------- + ------------------
\ 2 / / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \cos^{2}{\left(x \right)} + \cos^{2}{\left(x - \frac{\pi}{2} \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 1 10 (50 - 20*x)*log(x)
------- - ------- + ---------------- + ------------------
2 2 / 2 \ 2
csc (x) sec (x) x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{1}{\sec^{2}{\left(x \right)}} + \frac{1}{\csc^{2}{\left(x \right)}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2 2 10 (50 - 20*x)*log(x)
sin (x) - cos (x) + ---------------- + ------------------
/ 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 - cos(2*x) 1 + cos(2*x) 10 (50 - 20*x)*log(x)
------------ - ------------ + ---------------- + ------------------
2 2 / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{1 - \cos{\left(2 x \right)}}{2} + \frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{\cos{\left(2 x \right)} + 1}{2} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
1 1 10 (50 - 20*x)*log(x)
------------ - ------- + ---------------- + ------------------
2/ pi\ 2 / 2 \ 2
sec |x - --| sec (x) x*\1 + x - 5*x/ / 2 \
\ 2 / \1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{\sec^{2}{\left(x \right)}} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
10 (50 - 20*x)*log(x)
-cos(2*x) + ---------------- + ------------------
/ 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \cos{\left(2 x \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2
-1 + cot (x) 10 (50 - 20*x)*log(x)
- ------------ + ---------------- + ------------------
2 / 2 \ 2
1 + cot (x) x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
-(-1 + cot(x)^2)/(1 + cot(x)^2) + 10/(x*(1 + x^2 - 5*x)) + (50 - 20*x)*log(x)/(1 + x^2 - 5*x)^2
Potencias
[src]
2 2 10 (50 - 20*x)*log(x)
sin (x) - cos (x) + ---------------- + ------------------
/ 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} + \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
2 2
/ I*x -I*x\ / -I*x I*x\
|e e | \- e + e / 10 (50 - 20*x)*log(x)
- |---- + -----| - ----------------- + ---------------- + ------------------
\ 2 2 / 4 / 2 \ 2
x*\1 + x - 5*x/ / 2 \
\1 + x - 5*x/
$$\frac{\left(50 - 20 x\right) \log{\left(x \right)}}{\left(x^{2} - 5 x + 1\right)^{2}} - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2} - \frac{\left(e^{i x} - e^{- i x}\right)^{2}}{4} + \frac{10}{x \left(x^{2} - 5 x + 1\right)}$$
-(exp(i*x)/2 + exp(-i*x)/2)^2 - (-exp(-i*x) + exp(i*x))^2/4 + 10/(x*(1 + x^2 - 5*x)) + (50 - 20*x)*log(x)/(1 + x^2 - 5*x)^2