Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(((6*sin(x))/(1+cos(x))-2*sqrt(13)+4)/(4+2*sqrt(13)+(6*sin(x))/(1+cos(x))))/sqrt(13)
¿Cómo vas a descomponer esta log(((6*sin(x))/(1+cos(x))-2*sqrt(13)+4)/(4+2*sqrt(13)+(6*sin(x))/(1+cos(x))))/sqrt(13) expresión en fracciones?
Solución
Ha introducido
[src]
/ 6*sin(x) ____ \
|---------- - 2*\/ 13 + 4|
|1 + cos(x) |
log|-------------------------|
| ____ 6*sin(x) |
|4 + 2*\/ 13 + ----------|
\ 1 + cos(x)/
------------------------------
____
\/ 13
$$\frac{\log{\left(\frac{\left(- 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 4}{\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{\sqrt{13}}$$
log(((6*sin(x))/(1 + cos(x)) - 2*sqrt(13) + 4)/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))))/sqrt(13)
Simplificación general
[src]
/ / ____\\
____ |3*sin(x) + (1 + cos(x))*\2 - \/ 13 /|
\/ 13 *log|------------------------------------|
| / ____\|
\3*sin(x) + (1 + cos(x))*\2 + \/ 13 //
------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{\left(2 - \sqrt{13}\right) \left(\cos{\left(x \right)} + 1\right) + 3 \sin{\left(x \right)}}{\left(2 + \sqrt{13}\right) \left(\cos{\left(x \right)} + 1\right) + 3 \sin{\left(x \right)}} \right)}}{13}$$
sqrt(13)*log((3*sin(x) + (1 + cos(x))*(2 - sqrt(13)))/(3*sin(x) + (1 + cos(x))*(2 + sqrt(13))))/13
Descomposición de una fracción
[src]
sqrt(13)*log(4/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))) - 2*sqrt(13)/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))) + 6*sin(x)/(4 + 2*sqrt(13) + 4*cos(x) + (6*sin(x))/(1 + cos(x)) + 2*sqrt(13)*cos(x) + 6*cos(x)*sin(x)/(1 + cos(x))))/13
$$\frac{\sqrt{13} \log{\left(\frac{6 \sin{\left(x \right)}}{4 \cos{\left(x \right)} + 2 \sqrt{13} \cos{\left(x \right)} + 4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{2 \sqrt{13}}{\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{4}{\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
/ ____ \
____ | 4 2*\/ 13 6*sin(x) |
\/ 13 *log|------------------------- - ------------------------- + ------------------------------------------------------------------------|
| ____ 6*sin(x) ____ 6*sin(x) ____ 6*sin(x) ____ 6*cos(x)*sin(x)|
|4 + 2*\/ 13 + ---------- 4 + 2*\/ 13 + ---------- 4 + 2*\/ 13 + 4*cos(x) + ---------- + 2*\/ 13 *cos(x) + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
--------------------------------------------------------------------------------------------------------------------------------------------
13
Combinatoria
[src]
/ ____ \
____ | 4 2*\/ 13 6*sin(x) |
\/ 13 *log|------------------------- - ------------------------- + ------------------------------------------------------------------------|
| ____ 6*sin(x) ____ 6*sin(x) ____ ____ 6*sin(x) 6*cos(x)*sin(x)|
|4 + 2*\/ 13 + ---------- 4 + 2*\/ 13 + ---------- 4 + 2*\/ 13 + 4*cos(x) + 2*\/ 13 *cos(x) + ---------- + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
--------------------------------------------------------------------------------------------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{6 \sin{\left(x \right)}}{4 \cos{\left(x \right)} + 2 \sqrt{13} \cos{\left(x \right)} + 4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{2 \sqrt{13}}{4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{4}{4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
sqrt(13)*log(4/(4 + 2*sqrt(13) + 6*sin(x)/(1 + cos(x))) - 2*sqrt(13)/(4 + 2*sqrt(13) + 6*sin(x)/(1 + cos(x))) + 6*sin(x)/(4 + 2*sqrt(13) + 4*cos(x) + 2*sqrt(13)*cos(x) + 6*sin(x)/(1 + cos(x)) + 6*cos(x)*sin(x)/(1 + cos(x))))/13
Unión de expresiones racionales
[src]
/ ____ \
____ |2 + 2*cos(x) + 3*sin(x) - \/ 13 *(1 + cos(x))|
\/ 13 *log|---------------------------------------------|
| / ____\ |
\ 3*sin(x) + (1 + cos(x))*\2 + \/ 13 / /
---------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- \sqrt{13} \left(\cos{\left(x \right)} + 1\right) + 3 \sin{\left(x \right)} + 2 \cos{\left(x \right)} + 2}{\left(2 + \sqrt{13}\right) \left(\cos{\left(x \right)} + 1\right) + 3 \sin{\left(x \right)}} \right)}}{13}$$
sqrt(13)*log((2 + 2*cos(x) + 3*sin(x) - sqrt(13)*(1 + cos(x)))/(3*sin(x) + (1 + cos(x))*(2 + sqrt(13))))/13
Denominador común
[src]
/ ____ \
____ | 2 \/ 13 3*sin(x) |
\/ 13 *log|----------------------- - ----------------------- + --------------------------------------------------------------------|
| ____ 3*sin(x) ____ 3*sin(x) ____ ____ 3*sin(x) 3*cos(x)*sin(x)|
|2 + \/ 13 + ---------- 2 + \/ 13 + ---------- 2 + \/ 13 + 2*cos(x) + \/ 13 *cos(x) + ---------- + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
------------------------------------------------------------------------------------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{3 \sin{\left(x \right)}}{2 \cos{\left(x \right)} + \sqrt{13} \cos{\left(x \right)} + 2 + \sqrt{13} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{\sqrt{13}}{2 + \sqrt{13} + \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{2}{2 + \sqrt{13} + \frac{3 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
sqrt(13)*log(2/(2 + sqrt(13) + 3*sin(x)/(1 + cos(x))) - sqrt(13)/(2 + sqrt(13) + 3*sin(x)/(1 + cos(x))) + 3*sin(x)/(2 + sqrt(13) + 2*cos(x) + sqrt(13)*cos(x) + 3*sin(x)/(1 + cos(x)) + 3*cos(x)*sin(x)/(1 + cos(x))))/13
Potencias
[src]
/ / -I*x I*x\\
| ____ 3*I*\- e + e /|
|4 - 2*\/ 13 - --------------------|
| I*x -I*x |
| e e |
| 1 + ---- + ----- |
____ | 2 2 |
\/ 13 *log|-----------------------------------|
| / -I*x I*x\|
| ____ 3*I*\- e + e /|
|4 + 2*\/ 13 - --------------------|
| I*x -I*x |
| e e |
| 1 + ---- + ----- |
\ 2 2 /
-----------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- \frac{3 i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} - 2 \sqrt{13} + 4}{- \frac{3 i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} + 4 + 2 \sqrt{13}} \right)}}{13}$$
/ ____ 6*sin(x) \
|4 - 2*\/ 13 + ----------|
____ | 1 + cos(x)|
\/ 13 *log|-------------------------|
| ____ 6*sin(x) |
|4 + 2*\/ 13 + ----------|
\ 1 + cos(x)/
-------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
sqrt(13)*log((4 - 2*sqrt(13) + 6*sin(x)/(1 + cos(x)))/(4 + 2*sqrt(13) + 6*sin(x)/(1 + cos(x))))/13
Denominador racional
[src]
/ / ____ ____ / ____ 6*sin(x) ____ 6*cos(x)*sin(x)\ 2*6*sin(x) / ____ 6*sin(x) \ ____ 12*cos(x)*sin(x)\\
|2*|8 + 4*\/ 13 + 8*cos(x) - \/ 13 *|4 + 2*\/ 13 + 4*cos(x) + ---------- + 2*\/ 13 *cos(x) + ---------------| + ---------- + 3*|4 + 2*\/ 13 + ----------|*sin(x) + 4*\/ 13 *cos(x) + ----------------||
____ | \ \ 1 + cos(x) 1 + cos(x) / 1 + cos(x) \ 1 + cos(x)/ 1 + cos(x) /|
\/ 13 *log|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| / ____ 6*sin(x) \ / ____ 6*sin(x) ____ 6*cos(x)*sin(x)\ |
| |4 + 2*\/ 13 + ----------|*|4 + 2*\/ 13 + 4*cos(x) + ---------- + 2*\/ 13 *cos(x) + ---------------| |
\ \ 1 + cos(x)/ \ 1 + cos(x) 1 + cos(x) / /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{2 \left(3 \left(\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(x \right)} - \sqrt{13} \left(4 \cos{\left(x \right)} + 2 \sqrt{13} \cos{\left(x \right)} + 4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 8 \cos{\left(x \right)} + 4 \sqrt{13} \cos{\left(x \right)} + 8 + 4 \sqrt{13} + \frac{12 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{2 \cdot 6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)}{\left(\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \left(4 \cos{\left(x \right)} + 2 \sqrt{13} \cos{\left(x \right)} + 4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)} \right)}}{13}$$
sqrt(13)*log(2*(8 + 4*sqrt(13) + 8*cos(x) - sqrt(13)*(4 + 2*sqrt(13) + 4*cos(x) + (6*sin(x))/(1 + cos(x)) + 2*sqrt(13)*cos(x) + 6*cos(x)*sin(x)/(1 + cos(x))) + 2*(6*sin(x))/(1 + cos(x)) + 3*(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x)))*sin(x) + 4*sqrt(13)*cos(x) + 12*cos(x)*sin(x)/(1 + cos(x)))/((4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x)))*(4 + 2*sqrt(13) + 4*cos(x) + (6*sin(x))/(1 + cos(x)) + 2*sqrt(13)*cos(x) + 6*cos(x)*sin(x)/(1 + cos(x)))))/13
Respuesta numérica
[src]
0.277350098112615*log(((6*sin(x))/(1 + cos(x)) - 2*sqrt(13) + 4)/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))))
0.277350098112615*log(((6*sin(x))/(1 + cos(x)) - 2*sqrt(13) + 4)/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))))
Compilar la expresión
[src]
/ 6*sin(x) ____ \
|---------- - 2*\/ 13 + 4|
____ |1 + cos(x) |
\/ 13 *log|-------------------------|
| ____ 6*sin(x) |
|4 + 2*\/ 13 + ----------|
\ 1 + cos(x)/
-------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{\left(- 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 4}{\left(4 + 2 \sqrt{13}\right) + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
sqrt(13)*log(((6*sin(x))/(1 + cos(x)) - 2*sqrt(13) + 4)/(4 + 2*sqrt(13) + (6*sin(x))/(1 + cos(x))))/13
Parte trigonométrica
[src]
/ ____ 6*sin(x) \
|4 - 2*\/ 13 + ---------------|
| / pi\|
| 1 + sin|x + --||
____ | \ 2 /|
\/ 13 *log|------------------------------|
| ____ 6*sin(x) |
|4 + 2*\/ 13 + ---------------|
| / pi\|
| 1 + sin|x + --||
\ \ 2 //
------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}}{4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}} \right)}}{13}$$
/ / pi\\
| 6*cos|x - --||
| ____ \ 2 /|
|4 - 2*\/ 13 + -------------|
____ | 1 + cos(x) |
\/ 13 *log|----------------------------|
| / pi\|
| 6*cos|x - --||
| ____ \ 2 /|
|4 + 2*\/ 13 + -------------|
\ 1 + cos(x) /
----------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}}{4 + 2 \sqrt{13} + \frac{6 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
/ ____ 6 \
|4 - 2*\/ 13 + ------------------------|
| / 1 \ |
| |1 + -----------|*csc(x)|
| | /pi \| |
| | csc|-- - x|| |
____ | \ \2 // |
\/ 13 *log|---------------------------------------|
| ____ 6 |
|4 + 2*\/ 13 + ------------------------|
| / 1 \ |
| |1 + -----------|*csc(x)|
| | /pi \| |
| | csc|-- - x|| |
\ \ \2 // /
---------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}}{4 + 2 \sqrt{13} + \frac{6}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}} \right)}}{13}$$
/ ____ 6 \
|4 - 2*\/ 13 + ------------------------|
| / 1 \ / pi\|
| |1 + ------|*sec|x - --||
____ | \ sec(x)/ \ 2 /|
\/ 13 *log|---------------------------------------|
| ____ 6 |
|4 + 2*\/ 13 + ------------------------|
| / 1 \ / pi\|
| |1 + ------|*sec|x - --||
\ \ sec(x)/ \ 2 //
---------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}}{4 + 2 \sqrt{13} + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}} \right)}}{13}$$
/ /x\ \
| 12*cot|-| |
| ____ \2/ |
|4 - 2*\/ 13 + --------------------------------|
| / 2/x\\|
| | -1 + cot |-|||
| / 2/x\\ | \2/||
| |1 + cot |-||*|1 + ------------||
| \ \2// | 2/x\ ||
| | 1 + cot |-| ||
____ | \ \2/ /|
\/ 13 *log|-----------------------------------------------|
| /x\ |
| 12*cot|-| |
| ____ \2/ |
|4 + 2*\/ 13 + --------------------------------|
| / 2/x\\|
| | -1 + cot |-|||
| / 2/x\\ | \2/||
| |1 + cot |-||*|1 + ------------||
| \ \2// | 2/x\ ||
| | 1 + cot |-| ||
\ \ \2/ //
-----------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{12 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{4 + 2 \sqrt{13} + \frac{12 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}} \right)}}{13}$$
/ ____ 6 \
|4 - 2*\/ 13 + -------------------|
| / 1 \ |
| |1 + ------|*csc(x)|
____ | \ sec(x)/ |
\/ 13 *log|----------------------------------|
| ____ 6 |
|4 + 2*\/ 13 + -------------------|
| / 1 \ |
| |1 + ------|*csc(x)|
\ \ sec(x)/ /
----------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}}{4 + 2 \sqrt{13} + \frac{6}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}} \right)}}{13}$$
/ /x\ \
| 12*tan|-| |
| ____ \2/ |
|4 - 2*\/ 13 + -------------------------------|
| / 2/x\\|
| | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-||*|1 + -----------||
| \ \2// | 2/x\||
| | 1 + tan |-|||
____ | \ \2//|
\/ 13 *log|----------------------------------------------|
| /x\ |
| 12*tan|-| |
| ____ \2/ |
|4 + 2*\/ 13 + -------------------------------|
| / 2/x\\|
| | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-||*|1 + -----------||
| \ \2// | 2/x\||
| | 1 + tan |-|||
\ \ \2///
----------------------------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{12 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{4 + 2 \sqrt{13} + \frac{12 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}} \right)}}{13}$$
/ ____ 6*sin(x) \
|4 - 2*\/ 13 + ----------|
____ | 1 + cos(x)|
\/ 13 *log|-------------------------|
| ____ 6*sin(x) |
|4 + 2*\/ 13 + ----------|
\ 1 + cos(x)/
-------------------------------------
13
$$\frac{\sqrt{13} \log{\left(\frac{- 2 \sqrt{13} + 4 + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{4 + 2 \sqrt{13} + \frac{6 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{13}$$
sqrt(13)*log((4 - 2*sqrt(13) + 6*sin(x)/(1 + cos(x)))/(4 + 2*sqrt(13) + 6*sin(x)/(1 + cos(x))))/13