Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(tan(x)^2+1)/8-log(tan(x)+1)/4+(tan(x)-1)/(4*tan(x)^2+4)
¿Cómo vas a descomponer esta log(tan(x)^2+1)/8-log(tan(x)+1)/4+(tan(x)-1)/(4*tan(x)^2+4) expresión en fracciones?
Solución
Ha introducido
[src]
/ 2 \
log\tan (x) + 1/ log(tan(x) + 1) tan(x) - 1
---------------- - --------------- + -------------
8 4 2
4*tan (x) + 4
$$\left(- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}\right) + \frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4}$$
log(tan(x)^2 + 1)/8 - log(tan(x) + 1)/4 + (tan(x) - 1)/(4*tan(x)^2 + 4)
Simplificación general
[src]
/ 2 \ ___ / pi\
log|------------| \/ 2 *cos|2*x + --|
1 log(1 + tan(x)) \1 + cos(2*x)/ \ 4 /
- - - --------------- + ----------------- - -------------------
8 4 8 8
$$\frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8}$$
-1/8 - log(1 + tan(x))/4 + log(2/(1 + cos(2*x)))/8 - sqrt(2)*cos(2*x + pi/4)/8
Denominador racional
[src]
2 / 2 \ 2 / 2 \
-2 - 2*log(1 + tan(x)) + 2*tan(x) + tan (x)*log\1 + tan (x)/ - 2*tan (x)*log(1 + tan(x)) + log\1 + tan (x)/
-----------------------------------------------------------------------------------------------------------
2
8 + 8*tan (x)
$$\frac{- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} + 2 \tan{\left(x \right)} - 2}{8 \tan^{2}{\left(x \right)} + 8}$$
(-2 - 2*log(1 + tan(x)) + 2*tan(x) + tan(x)^2*log(1 + tan(x)^2) - 2*tan(x)^2*log(1 + tan(x)) + log(1 + tan(x)^2))/(8 + 8*tan(x)^2)
Respuesta numérica
[src]
0.125*log(tan(x)^2 + 1) - 0.25*log(tan(x) + 1) + (-1.0 + tan(x))/(4.0 + 4.0*tan(x)^2)
0.125*log(tan(x)^2 + 1) - 0.25*log(tan(x) + 1) + (-1.0 + tan(x))/(4.0 + 4.0*tan(x)^2)
Denominador común
[src]
/ 2 \
log(1 + tan(x)) log\1 + tan (x)/ -1 + tan(x)
- --------------- + ---------------- + -------------
4 8 2
4 + 4*tan (x)
$$\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}$$
-log(1 + tan(x))/4 + log(1 + tan(x)^2)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Unión de expresiones racionales
[src]
/ 2 \ / / 2 \\
-2 + 2*tan(x) + \1 + tan (x)/*\-2*log(1 + tan(x)) + log\1 + tan (x)//
---------------------------------------------------------------------
/ 2 \
8*\1 + tan (x)/
$$\frac{\left(- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}\right) \left(\tan^{2}{\left(x \right)} + 1\right) + 2 \tan{\left(x \right)} - 2}{8 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
(-2 + 2*tan(x) + (1 + tan(x)^2)*(-2*log(1 + tan(x)) + log(1 + tan(x)^2)))/(8*(1 + tan(x)^2))
Potencias
[src]
/ 2\
| / I*x -I*x\ |
/ / I*x -I*x\\ | \- e + e / | / I*x -I*x\
| I*\- e + e /| log|1 - -----------------| I*\- e + e /
log|1 + ------------------| | 2 | -1 + ------------------
| I*x -I*x | | / I*x -I*x\ | I*x -I*x
\ e + e / \ \e + e / / e + e
- --------------------------- + -------------------------- + -----------------------
4 8 2
/ I*x -I*x\
4*\- e + e /
4 - -------------------
2
/ I*x -I*x\
\e + e /
$$\frac{\log{\left(- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 1 \right)}}{8} - \frac{\log{\left(\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 1 \right)}}{4} + \frac{\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} - 1}{- \frac{4 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 4}$$
/ 2 \
log(1 + tan(x)) log\1 + tan (x)/ -1 + tan(x)
- --------------- + ---------------- + -------------
4 8 2
4 + 4*tan (x)
$$\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}$$
-log(1 + tan(x))/4 + log(1 + tan(x)^2)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Compilar la expresión
[src]
/ 2 \
log(tan(x) + 1) log\tan (x) + 1/ -1 + tan(x)
- --------------- + ---------------- + -------------
4 8 2
4 + 4*tan (x)
$$\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}$$
-log(tan(x) + 1)/4 + log(tan(x)^2 + 1)/8 + (-1 + tan(x))/(4 + 4*tan(x)^2)
Combinatoria
[src]
/ / 2 \ 2 / 2 \ 2 \
-\2 - log\1 + tan (x)/ - 2*tan(x) + 2*log(1 + tan(x)) - tan (x)*log\1 + tan (x)/ + 2*tan (x)*log(1 + tan(x))/
--------------------------------------------------------------------------------------------------------------
/ 2 \
8*\1 + tan (x)/
$$- \frac{2 \log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} + 2 \log{\left(\tan{\left(x \right)} + 1 \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)} - \log{\left(\tan^{2}{\left(x \right)} + 1 \right)} - 2 \tan{\left(x \right)} + 2}{8 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
-(2 - log(1 + tan(x)^2) - 2*tan(x) + 2*log(1 + tan(x)) - tan(x)^2*log(1 + tan(x)^2) + 2*tan(x)^2*log(1 + tan(x)))/(8*(1 + tan(x)^2))
Parte trigonométrica
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/ sec(x) \
log|1 + -----------|
| / pi\|
| sec|x - --|| / 2 \ ___
1 \ \ 2 // log\sec (x)/ \/ 2
- - - -------------------- + ------------ - ---------------
8 4 8 / pi\
8*sec|2*x + --|
\ 4 /
$$- \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{8} - \frac{1}{8} - \frac{\sqrt{2}}{8 \sec{\left(2 x + \frac{\pi}{4} \right)}}$$
/ 2 \
| sec (x)|
/ sec(x)\ log|1 + -------| sec(x)
log|1 + ------| | 2 | -1 + ------
\ csc(x)/ \ csc (x)/ csc(x)
- --------------- + ---------------- + -------------
4 8 2
4*sec (x)
4 + ---------
2
csc (x)
$$\frac{-1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}}{4 + \frac{4 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}} \right)}}{4}$$
/ 2 \
log(1 + tan(x)) log\1 + tan (x)/ -1 + tan(x)
- --------------- + ---------------- + -------------
4 8 2
4 + 4*tan (x)
$$\frac{\tan{\left(x \right)} - 1}{4 \tan^{2}{\left(x \right)} + 4} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{8}$$
/ 2 \
/ sec(x) \ | sec (x) | sec(x)
log|1 + -----------| log|1 + ------------| -1 + -----------
| / pi\| | 2/ pi\| / pi\
| sec|x - --|| | sec |x - --|| sec|x - --|
\ \ 2 // \ \ 2 // \ 2 /
- -------------------- + --------------------- + ----------------
4 8 2
4*sec (x)
4 + ------------
2/ pi\
sec |x - --|
\ 2 /
$$\frac{\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{4 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 4} - \frac{\log{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1 \right)}}{8}$$
/ 2\
|/ 2/x\\ |
||1 + tan |-|| |
|\ \2// |
log|--------------|
| 2|
|/ 2/x\\ | ___ / 2/ pi\\
||1 - tan |-|| | \/ 2 *|1 - tan |x + --||
1 log(1 + tan(x)) \\ \2// / \ \ 8 //
- - - --------------- + ------------------- - ------------------------
8 4 8 / 2/ pi\\
8*|1 + tan |x + --||
\ \ 8 //
$$- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{8 \left(\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \frac{\log{\left(\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8}$$
/ 2/ pi\\
/ / pi\\ | cos |x - --|| / pi\
| cos|x - --|| | \ 2 /| cos|x - --|
| \ 2 /| log|1 + ------------| \ 2 /
log|1 + -----------| | 2 | -1 + -----------
\ cos(x) / \ cos (x) / cos(x)
- -------------------- + --------------------- + ------------------
4 8 2/ pi\
4*cos |x - --|
\ 2 /
4 + --------------
2
cos (x)
$$\frac{-1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}}{4 + \frac{4 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{4}$$
___ / pi\
/ 2 \ \/ 2 *cos|2*x + --|
1 log(1 + tan(x)) log\sec (x)/ \ 4 /
- - - --------------- + ------------ - -------------------
8 4 8 8
$$- \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{8} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8}$$
/ 1 \
___ / pi\ -2*log(1 + tan(x)) + log|-------|
1 + \/ 2 *cos|2*x + --| | 2 |
\ 4 / \cos (x)/
- ----------------------- + ---------------------------------
8 8
$$- \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1}{8} + \frac{- 2 \log{\left(\tan{\left(x \right)} + 1 \right)} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8}$$
/ 1 \
___ / pi\ log|-------|
1 + \/ 2 *cos|2*x + --| | 2 |
log(1 + tan(x)) \ 4 / \cos (x)/
- --------------- - ----------------------- + ------------
4 8 8
$$- \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)} + 1}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8}$$
/ 2 \
| sin (x)|
/ sin(x)\ log|1 + -------| sin(x)
log|1 + ------| | 2 | -1 + ------
\ cos(x)/ \ cos (x)/ cos(x)
- --------------- + ---------------- + -------------
4 8 2
4*sin (x)
4 + ---------
2
cos (x)
$$\frac{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} - 1}{\frac{4 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 4} - \frac{\log{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1 \right)}}{8}$$
/ 2\
| / 2/x\\ |
| |1 + cot |-|| |
| \ \2// |
log|---------------|
| 2|
/ 1 \ |/ 2/x\\ | ___ / 2/ pi\\
log|1 + ------| ||-1 + cot |-|| | \/ 2 *|-1 + cot |x + --||
1 \ cot(x)/ \\ \2// / \ \ 8 //
- - - --------------- + -------------------- - -------------------------
8 4 8 / 2/ pi\\
8*|1 + cot |x + --||
\ \ 8 //
$$- \frac{\sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right)}{8 \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \frac{\log{\left(\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}} \right)}}{8} - \frac{\log{\left(1 + \frac{1}{\cot{\left(x \right)}} \right)}}{4} - \frac{1}{8}$$
/ 1 \
/ 1 \ log|1 + -------| 1
log|1 + ------| | 2 | -1 + ------
\ cot(x)/ \ cot (x)/ cot(x)
- --------------- + ---------------- + -----------
4 8 4
4 + -------
2
cot (x)
$$\frac{-1 + \frac{1}{\cot{\left(x \right)}}}{4 + \frac{4}{\cot^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{1}{\cot{\left(x \right)}} \right)}}{4}$$
/ 4 \
/ 2 \ | 4*sin (x)| 2
| 2*sin (x)| log|1 + ---------| 2*sin (x)
log|1 + ---------| | 2 | -1 + ---------
\ sin(2*x)/ \ sin (2*x)/ sin(2*x)
- ------------------ + ------------------ + --------------
4 8 4
16*sin (x)
4 + ----------
2
sin (2*x)
$$\frac{\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - 1}{\frac{16 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 4} - \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1 \right)}}{8}$$
/ 2 \ ___ / 2/ pi\\
log|------------| \/ 2 *|1 - tan |x + --||
1 log(1 + tan(x)) \1 + cos(2*x)/ \ \ 8 //
- - - --------------- + ----------------- - ------------------------
8 4 8 / 2/ pi\\
8*|1 + tan |x + --||
\ \ 8 //
$$- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(x + \frac{\pi}{8} \right)}\right)}{8 \left(\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1\right)} + \frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8}$$
/ 2 \ ___ / pi\
log|------------| \/ 2 *cos|2*x + --|
1 log(1 + tan(x)) \1 + cos(2*x)/ \ 4 /
- - - --------------- + ----------------- - -------------------
8 4 8 8
$$\frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8}$$
/ / pi\\
| cos|x - --|| / 1 \
| \ 2 /| log|-------| ___ / pi\
log|1 + -----------| | 2 | \/ 2 *cos|2*x + --|
1 \ cos(x) / \cos (x)/ \ 4 /
- - - -------------------- + ------------ - -------------------
8 4 8 8
$$- \frac{\log{\left(1 + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{8} - \frac{\sqrt{2} \cos{\left(2 x + \frac{\pi}{4} \right)}}{8} - \frac{1}{8}$$
/ 2/pi \\
/ /pi \\ | csc |-- - x|| /pi \
| csc|-- - x|| | \2 /| csc|-- - x|
| \2 /| log|1 + ------------| \2 /
log|1 + -----------| | 2 | -1 + -----------
\ csc(x) / \ csc (x) / csc(x)
- -------------------- + --------------------- + ------------------
4 8 2/pi \
4*csc |-- - x|
\2 /
4 + --------------
2
csc (x)
$$\frac{-1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}}{4 + \frac{4 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}} + \frac{\log{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}} \right)}}{8} - \frac{\log{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{4}$$
/ 1 \
/ 2 \ log|------------|
| 2*sin (x)| | 2/ pi\| ___ / 3*pi\
log|1 + ---------| |sin |x + --|| \/ 2 *sin|2*x + ----|
1 \ sin(2*x)/ \ \ 2 // \ 4 /
- - - ------------------ + ----------------- - ---------------------
8 4 8 8
$$- \frac{\log{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + 1 \right)}}{4} + \frac{\log{\left(\frac{1}{\sin^{2}{\left(x + \frac{\pi}{2} \right)}} \right)}}{8} - \frac{\sqrt{2} \sin{\left(2 x + \frac{3 \pi}{4} \right)}}{8} - \frac{1}{8}$$
/ /pi \\
| csc|-- - x||
| \2 /| / 2/pi \\
log|1 + -----------| log|csc |-- - x|| ___
1 \ csc(x) / \ \2 // \/ 2
- - - -------------------- + ----------------- - ----------------
8 4 8 / pi\
8*csc|-2*x + --|
\ 4 /
$$- \frac{\log{\left(1 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} \right)}}{4} + \frac{\log{\left(\csc^{2}{\left(- x + \frac{\pi}{2} \right)} \right)}}{8} - \frac{1}{8} - \frac{\sqrt{2}}{8 \csc{\left(- 2 x + \frac{\pi}{4} \right)}}$$
/ 2 \ ___ 2/ pi\ / 2/ pi\\
log|------------| \/ 2 *sin |x + --|*|-1 + cot |x + --||
1 log(1 + tan(x)) \1 + cos(2*x)/ \ 8 / \ \ 8 //
- - - --------------- + ----------------- - --------------------------------------
8 4 8 8
$$- \frac{\sqrt{2} \left(\cot^{2}{\left(x + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(x + \frac{\pi}{8} \right)}}{8} + \frac{\log{\left(\frac{2}{\cos{\left(2 x \right)} + 1} \right)}}{8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{4} - \frac{1}{8}$$
-1/8 - log(1 + tan(x))/4 + log(2/(1 + cos(2*x)))/8 - sqrt(2)*sin(x + pi/8)^2*(-1 + cot(x + pi/8)^2)/8