Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(x)^atan(x+pi/4)*(log(cos(x))/(1+(x+pi/4)^2)-atan(x+pi/4)*sin(x)/cos(x))
¿Cómo vas a descomponer esta cos(x)^atan(x+pi/4)*(log(cos(x))/(1+(x+pi/4)^2)-atan(x+pi/4)*sin(x)/cos(x)) expresión en fracciones?
Solución
Ha introducido
[src]
/ pi\ / / pi\ \
atan|x + --| | atan|x + --|*sin(x)|
\ 4 / | log(cos(x)) \ 4 / |
(cos(x)) *|------------- - -------------------|
| 2 cos(x) |
| / pi\ |
|1 + |x + --| |
\ \ 4 / /
$$\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)}$$
cos(x)^atan(x + pi/4)*(log(cos(x))/(1 + (x + pi/4)^2) - atan(x + pi/4)*sin(x)/cos(x))
Simplificación general
[src]
/ pi\
-1 + atan|x + --|
\ 4 / / / 2\ / pi\ \
(cos(x)) *|16*cos(x)*log(cos(x)) - \16 + (pi + 4*x) /*atan|x + --|*sin(x)|
\ \ 4 / /
------------------------------------------------------------------------------------------
2
16 + (pi + 4*x)
$$\frac{\left(- \left(\left(4 x + \pi\right)^{2} + 16\right) \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + 16 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)} - 1}{\left(x \right)}}{\left(4 x + \pi\right)^{2} + 16}$$
cos(x)^(-1 + atan(x + pi/4))*(16*cos(x)*log(cos(x)) - (16 + (pi + 4*x)^2)*atan(x + pi/4)*sin(x))/(16 + (pi + 4*x)^2)
Denominador racional
[src]
/ pi\
-1 + atan|x + --|
\ 4 / / / pi\ 2 / pi\ \
-(cos(x)) *|-16*cos(x)*log(cos(x)) + 16*atan|x + --|*sin(x) + (pi + 4*x) *atan|x + --|*sin(x)|
\ \ 4 / \ 4 / /
---------------------------------------------------------------------------------------------------------------
2 2
16 + pi + 16*x + 8*pi*x
$$- \frac{\left(\left(4 x + \pi\right)^{2} \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} - 16 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + 16 \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)} - 1}{\left(x \right)}}{16 x^{2} + 8 \pi x + \pi^{2} + 16}$$
-cos(x)^(-1 + atan(x + pi/4))*(-16*cos(x)*log(cos(x)) + 16*atan(x + pi/4)*sin(x) + (pi + 4*x)^2*atan(x + pi/4)*sin(x))/(16 + pi^2 + 16*x^2 + 8*pi*x)
Denominador común
[src]
/ / pi\ / pi\ / pi\ / pi\ / pi\ \
| atan|x + --| atan|x + --| atan|x + --| atan|x + --| atan|x + --| |
| \ 4 / \ 4 / / pi\ 2 \ 4 / / pi\ 2 \ 4 / / pi\ \ 4 / / pi\ |
-|- 16*(cos(x)) *cos(x)*log(cos(x)) + 16*(cos(x)) *atan|x + --|*sin(x) + pi *(cos(x)) *atan|x + --|*sin(x) + 16*x *(cos(x)) *atan|x + --|*sin(x) + 8*pi*x*(cos(x)) *atan|x + --|*sin(x)|
\ \ 4 / \ 4 / \ 4 / \ 4 / /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2
16*cos(x) + pi *cos(x) + 16*x *cos(x) + 8*pi*x*cos(x)
$$- \frac{16 x^{2} \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + 8 \pi x \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} - 16 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)} + \pi^{2} \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + 16 \sin{\left(x \right)} \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{16 x^{2} \cos{\left(x \right)} + 8 \pi x \cos{\left(x \right)} + \pi^{2} \cos{\left(x \right)} + 16 \cos{\left(x \right)}}$$
-(-16*cos(x)^atan(x + pi/4)*cos(x)*log(cos(x)) + 16*cos(x)^atan(x + pi/4)*atan(x + pi/4)*sin(x) + pi^2*cos(x)^atan(x + pi/4)*atan(x + pi/4)*sin(x) + 16*x^2*cos(x)^atan(x + pi/4)*atan(x + pi/4)*sin(x) + 8*pi*x*cos(x)^atan(x + pi/4)*atan(x + pi/4)*sin(x))/(16*cos(x) + pi^2*cos(x) + 16*x^2*cos(x) + 8*pi*x*cos(x))
Combinatoria
[src]
/ pi\
atan|x + --|
\ 4 / / / pi\ 2 / pi\ 2 / pi\ / pi\ \
-(cos(x)) *|-16*cos(x)*log(cos(x)) + 16*atan|x + --|*sin(x) + pi *atan|x + --|*sin(x) + 16*x *atan|x + --|*sin(x) + 8*pi*x*atan|x + --|*sin(x)|
\ \ 4 / \ 4 / \ 4 / \ 4 / /
-----------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2 2 \
\16 + pi + 16*x + 8*pi*x/*cos(x)
$$- \frac{\left(16 x^{2} \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + 8 \pi x \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} - 16 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + \pi^{2} \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + 16 \sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)}}{\left(16 x^{2} + 8 \pi x + \pi^{2} + 16\right) \cos{\left(x \right)}}$$
-cos(x)^atan(x + pi/4)*(-16*cos(x)*log(cos(x)) + 16*atan(x + pi/4)*sin(x) + pi^2*atan(x + pi/4)*sin(x) + 16*x^2*atan(x + pi/4)*sin(x) + 8*pi*x*atan(x + pi/4)*sin(x))/((16 + pi^2 + 16*x^2 + 8*pi*x)*cos(x))
Potencias
[src]
/ pi\
atan|x + --| / / I*x -I*x\ \
\ 4 / | |e e | / -I*x I*x\ / pi\|
/ I*x -I*x\ |log|---- + -----| I*\- e + e /*atan|x + --||
|e e | | \ 2 2 / \ 4 /|
|---- + -----| *|----------------- + -------------------------------|
\ 2 2 / | 2 / I*x -I*x\ |
| / pi\ |e e | |
| 1 + |x + --| 2*|---- + -----| |
\ \ 4 / \ 2 2 / /
$$\left(\frac{i \left(e^{i x} - e^{- i x}\right) \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{2 \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)} + \frac{\log{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}$$
(exp(i*x)/2 + exp(-i*x)/2)^atan(x + pi/4)*(log(exp(i*x)/2 + exp(-i*x)/2)/(1 + (x + pi/4)^2) + i*(-exp(-i*x) + exp(i*x))*atan(x + pi/4)/(2*(exp(i*x)/2 + exp(-i*x)/2)))
Unión de expresiones racionales
[src]
/pi + 4*x\
atan|--------|
\ 4 / / / 2\ /pi + 4*x\ \
(cos(x)) *|16*cos(x)*log(cos(x)) - \16 + (pi + 4*x) /*atan|--------|*sin(x)|
\ \ 4 / /
-----------------------------------------------------------------------------------------
/ 2\
\16 + (pi + 4*x) /*cos(x)
$$\frac{\left(- \left(\left(4 x + \pi\right)^{2} + 16\right) \sin{\left(x \right)} \operatorname{atan}{\left(\frac{4 x + \pi}{4} \right)} + 16 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}\right) \cos^{\operatorname{atan}{\left(\frac{4 x + \pi}{4} \right)}}{\left(x \right)}}{\left(\left(4 x + \pi\right)^{2} + 16\right) \cos{\left(x \right)}}$$
cos(x)^atan((pi + 4*x)/4)*(16*cos(x)*log(cos(x)) - (16 + (pi + 4*x)^2)*atan((pi + 4*x)/4)*sin(x))/((16 + (pi + 4*x)^2)*cos(x))
Parte trigonométrica
[src]
/ / 2/x\\ \
/ pi\ | |-1 + cot |-|| |
atan|x + --| | | \2/| |
\ 4 / |log|------------| |
/ 2/x\\ | | 2/x\ | / pi\|
|-1 + cot |-|| | |1 + cot |-| | atan|x + --||
| \2/| | \ \2/ / \ 4 /|
|------------| *|----------------- - ------------|
| 2/x\ | | 2 cot(x) |
|1 + cot |-| | | / pi\ |
\ \2/ / | 1 + |x + --| |
\ \ 4 / /
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}} \left(- \frac{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\cot{\left(x \right)}} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right)$$
/ / 2/x\\ \
/ pi\ | |1 - tan |-|| |
atan|x + --| | | \2/| |
\ 4 / |log|-----------| |
/ 2/x\\ | | 2/x\| |
|1 - tan |-|| | |1 + tan |-|| |
| \2/| | \ \2// / pi\ |
|-----------| *|---------------- - atan|x + --|*tan(x)|
| 2/x\| | 2 \ 4 / |
|1 + tan |-|| | / pi\ |
\ \2// | 1 + |x + --| |
\ \ 4 / /
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}} \left(- \tan{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + \frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right)$$
/ pi\
atan|x + --| / / / pi\\ 2 / pi\\
\ 4 / |log|sin|x + --|| 2*sin (x)*atan|x + --||
/ / pi\\ | \ \ 2 // \ 4 /|
|sin|x + --|| *|---------------- - ----------------------|
\ \ 2 // | 2 sin(2*x) |
| / pi\ |
| 1 + |x + --| |
\ \ 4 / /
$$\left(- \frac{2 \sin^{2}{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(2 x \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x + \frac{\pi}{2} \right)}$$
/ pi\
atan|x + --| / / 1 \ / pi\ \
\ 4 / | log|------| atan|x + --|*sec(x)|
/ 1 \ | \sec(x)/ \ 4 / |
|------| *|------------- - -------------------|
\sec(x)/ | 2 csc(x) |
| / pi\ |
|1 + |x + --| |
\ \ 4 / /
$$\left(- \frac{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}$$
/ / 1 \ \
/ pi\ |log|-----------| |
atan|x + --| | | /pi \| / pi\ /pi \|
\ 4 / | |csc|-- - x|| atan|x + --|*csc|-- - x||
/ 1 \ | \ \2 // \ 4 / \2 /|
|-----------| *|---------------- - ------------------------|
| /pi \| | 2 csc(x) |
|csc|-- - x|| | / pi\ |
\ \2 // | 1 + |x + --| |
\ \ 4 / /
$$\left(- \frac{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{\log{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}$$
/ pi\
atan|x + --| / / / pi\\ / pi\ \
\ 4 / |log|sin|x + --|| atan|x + --|*sin(x)|
/ / pi\\ | \ \ 2 // \ 4 / |
|sin|x + --|| *|---------------- - -------------------|
\ \ 2 // | 2 / pi\ |
| / pi\ sin|x + --| |
| 1 + |x + --| \ 2 / |
\ \ 4 / /
$$\left(- \frac{\sin{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\sin{\left(x + \frac{\pi}{2} \right)}} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \sin^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x + \frac{\pi}{2} \right)}$$
/ / 2/x\\ \
/ pi\ | |1 - tan |-|| |
atan|x + --| | | \2/| |
\ 4 / |log|-----------| |
/ 2/x\\ | | 2/x\| / pi\ /x\|
|1 - tan |-|| | |1 + tan |-|| 2*atan|x + --|*tan|-||
| \2/| | \ \2// \ 4 / \2/|
|-----------| *|---------------- - ---------------------|
| 2/x\| | 2 2/x\ |
|1 + tan |-|| | / pi\ 1 - tan |-| |
\ \2// | 1 + |x + --| \2/ |
\ \ 4 / /
$$\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}} \left(\frac{\log{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1} - \frac{2 \tan{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{1 - \tan^{2}{\left(\frac{x}{2} \right)}}\right)$$
/ pi\ / / pi\ / pi\\
atan|x + --| | atan|x + --|*cos|x - --||
\ 4 / | log(cos(x)) \ 4 / \ 2 /|
(cos(x)) *|------------- - ------------------------|
| 2 cos(x) |
| / pi\ |
|1 + |x + --| |
\ \ 4 / /
$$\left(- \frac{\cos{\left(x - \frac{\pi}{2} \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\cos{\left(x \right)}} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)}$$
/ pi\
atan|x + --| / / 1 \ / pi\ \
\ 4 / | log|------| atan|x + --|*sec(x)|
/ 1 \ | \sec(x)/ \ 4 / |
|------| *|------------- - -------------------|
\sec(x)/ | 2 / pi\ |
| / pi\ sec|x - --| |
|1 + |x + --| \ 2 / |
\ \ 4 / /
$$\left(- \frac{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{\log{\left(\frac{1}{\sec{\left(x \right)}} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \left(\frac{1}{\sec{\left(x \right)}}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}$$
/ / 2/x\\ \
/ pi\ | |-1 + cot |-|| |
atan|x + --| | | \2/| |
\ 4 / |log|------------| |
/ 2/x\\ | | 2/x\ | / pi\ /x\|
|-1 + cot |-|| | |1 + cot |-| | 2*atan|x + --|*cot|-||
| \2/| | \ \2/ / \ 4 / \2/|
|------------| *|----------------- - ---------------------|
| 2/x\ | | 2 2/x\ |
|1 + cot |-| | | / pi\ -1 + cot |-| |
\ \2/ / | 1 + |x + --| \2/ |
\ \ 4 / /
$$\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}\right)^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}} \left(- \frac{2 \cot{\left(\frac{x}{2} \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right)$$
/ pi\
atan|x + --|
\ 4 / / log(cos(x)) / pi\ \
(cos(x)) *|------------- - atan|x + --|*tan(x)|
| 2 \ 4 / |
| / pi\ |
|1 + |x + --| |
\ \ 4 / /
$$\left(- \tan{\left(x \right)} \operatorname{atan}{\left(x + \frac{\pi}{4} \right)} + \frac{\log{\left(\cos{\left(x \right)} \right)}}{\left(x + \frac{\pi}{4}\right)^{2} + 1}\right) \cos^{\operatorname{atan}{\left(x + \frac{\pi}{4} \right)}}{\left(x \right)}$$
cos(x)^atan(x + pi/4)*(log(cos(x))/(1 + (x + pi/4)^2) - atan(x + pi/4)*tan(x))
Respuesta numérica
[src]
cos(x)^atan(x + pi/4)*(log(cos(x))/(1.0 + (0.785398163397448 + x)^2) - atan(x + pi/4)*sin(x)/cos(x))
cos(x)^atan(x + pi/4)*(log(cos(x))/(1.0 + (0.785398163397448 + x)^2) - atan(x + pi/4)*sin(x)/cos(x))