Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
log(((4*sin(x))/(1+cos(x))-sqrt(41)+5)/(5+sqrt(41)+(4*sin(x))/(1+cos(x))))/sqrt(41)
¿Cómo vas a descomponer esta log(((4*sin(x))/(1+cos(x))-sqrt(41)+5)/(5+sqrt(41)+(4*sin(x))/(1+cos(x))))/sqrt(41) expresión en fracciones?
Solución
Ha introducido
[src]
/ 4*sin(x) ____ \
|---------- - \/ 41 + 5|
|1 + cos(x) |
log|-----------------------|
| ____ 4*sin(x) |
|5 + \/ 41 + ----------|
\ 1 + cos(x)/
----------------------------
____
\/ 41
$$\frac{\log{\left(\frac{\left(- \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 5}{\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{\sqrt{41}}$$
log(((4*sin(x))/(1 + cos(x)) - sqrt(41) + 5)/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))))/sqrt(41)
Descomposición de una fracción
[src]
sqrt(41)*log(5/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))) - sqrt(41)/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))) + 4*sin(x)/(5 + sqrt(41) + 5*cos(x) + sqrt(41)*cos(x) + (4*sin(x))/(1 + cos(x)) + 4*cos(x)*sin(x)/(1 + cos(x))))/41
$$\frac{\sqrt{41} \log{\left(\frac{4 \sin{\left(x \right)}}{5 \cos{\left(x \right)} + \sqrt{41} \cos{\left(x \right)} + 5 + \sqrt{41} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{\sqrt{41}}{\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{5}{\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
/ ____ \
____ | 5 \/ 41 4*sin(x) |
\/ 41 *log|----------------------- - ----------------------- + --------------------------------------------------------------------|
| ____ 4*sin(x) ____ 4*sin(x) ____ ____ 4*sin(x) 4*cos(x)*sin(x)|
|5 + \/ 41 + ---------- 5 + \/ 41 + ---------- 5 + \/ 41 + 5*cos(x) + \/ 41 *cos(x) + ---------- + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
------------------------------------------------------------------------------------------------------------------------------------
41
Simplificación general
[src]
/ / ____\\
____ |4*sin(x) + (1 + cos(x))*\5 - \/ 41 /|
\/ 41 *log|------------------------------------|
| / ____\|
\4*sin(x) + (1 + cos(x))*\5 + \/ 41 //
------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{\left(5 - \sqrt{41}\right) \left(\cos{\left(x \right)} + 1\right) + 4 \sin{\left(x \right)}}{\left(5 + \sqrt{41}\right) \left(\cos{\left(x \right)} + 1\right) + 4 \sin{\left(x \right)}} \right)}}{41}$$
sqrt(41)*log((4*sin(x) + (1 + cos(x))*(5 - sqrt(41)))/(4*sin(x) + (1 + cos(x))*(5 + sqrt(41))))/41
Respuesta numérica
[src]
0.156173761888606*log(((4*sin(x))/(1 + cos(x)) - sqrt(41) + 5)/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))))
0.156173761888606*log(((4*sin(x))/(1 + cos(x)) - sqrt(41) + 5)/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))))
Potencias
[src]
/ ____ 4*sin(x) \
|5 - \/ 41 + ----------|
____ | 1 + cos(x)|
\/ 41 *log|-----------------------|
| ____ 4*sin(x) |
|5 + \/ 41 + ----------|
\ 1 + cos(x)/
-----------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
/ / -I*x I*x\\
| ____ 2*I*\- e + e /|
|5 - \/ 41 - --------------------|
| I*x -I*x |
| e e |
| 1 + ---- + ----- |
____ | 2 2 |
\/ 41 *log|---------------------------------|
| / -I*x I*x\|
| ____ 2*I*\- e + e /|
|5 + \/ 41 - --------------------|
| I*x -I*x |
| e e |
| 1 + ---- + ----- |
\ 2 2 /
---------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \frac{2 i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} - \sqrt{41} + 5}{- \frac{2 i \left(e^{i x} - e^{- i x}\right)}{\frac{e^{i x}}{2} + 1 + \frac{e^{- i x}}{2}} + 5 + \sqrt{41}} \right)}}{41}$$
sqrt(41)*log((5 - sqrt(41) - 2*i*(-exp(-i*x) + exp(i*x))/(1 + exp(i*x)/2 + exp(-i*x)/2))/(5 + sqrt(41) - 2*i*(-exp(-i*x) + exp(i*x))/(1 + exp(i*x)/2 + exp(-i*x)/2)))/41
Denominador común
[src]
/ ____ \
____ | 5 \/ 41 4*sin(x) |
\/ 41 *log|----------------------- - ----------------------- + --------------------------------------------------------------------|
| ____ 4*sin(x) ____ 4*sin(x) ____ ____ 4*sin(x) 4*cos(x)*sin(x)|
|5 + \/ 41 + ---------- 5 + \/ 41 + ---------- 5 + \/ 41 + 5*cos(x) + \/ 41 *cos(x) + ---------- + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
------------------------------------------------------------------------------------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{4 \sin{\left(x \right)}}{5 \cos{\left(x \right)} + \sqrt{41} \cos{\left(x \right)} + 5 + \sqrt{41} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{\sqrt{41}}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{5}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
sqrt(41)*log(5/(5 + sqrt(41) + 4*sin(x)/(1 + cos(x))) - sqrt(41)/(5 + sqrt(41) + 4*sin(x)/(1 + cos(x))) + 4*sin(x)/(5 + sqrt(41) + 5*cos(x) + sqrt(41)*cos(x) + 4*sin(x)/(1 + cos(x)) + 4*cos(x)*sin(x)/(1 + cos(x))))/41
Denominador racional
[src]
/ ____ ____ / ____ ____ 4*sin(x) 4*cos(x)*sin(x)\ / ____ 4*sin(x) \ ____ 5*4*sin(x) 20*cos(x)*sin(x)\
|25 + 5*\/ 41 + 25*cos(x) - \/ 41 *|5 + \/ 41 + 5*cos(x) + \/ 41 *cos(x) + ---------- + ---------------| + 4*|5 + \/ 41 + ----------|*sin(x) + 5*\/ 41 *cos(x) + ---------- + ----------------|
____ | \ 1 + cos(x) 1 + cos(x) / \ 1 + cos(x)/ 1 + cos(x) 1 + cos(x) |
\/ 41 *log|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| / ____ 4*sin(x) \ / ____ ____ 4*sin(x) 4*cos(x)*sin(x)\ |
| |5 + \/ 41 + ----------|*|5 + \/ 41 + 5*cos(x) + \/ 41 *cos(x) + ---------- + ---------------| |
\ \ 1 + cos(x)/ \ 1 + cos(x) 1 + cos(x) / /
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{4 \left(\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \sin{\left(x \right)} - \sqrt{41} \left(5 \cos{\left(x \right)} + \sqrt{41} \cos{\left(x \right)} + 5 + \sqrt{41} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 25 \cos{\left(x \right)} + 5 \sqrt{41} \cos{\left(x \right)} + 25 + 5 \sqrt{41} + \frac{20 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{5 \cdot 4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{\left(\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) \left(5 \cos{\left(x \right)} + \sqrt{41} \cos{\left(x \right)} + 5 + \sqrt{41} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right)} \right)}}{41}$$
sqrt(41)*log((25 + 5*sqrt(41) + 25*cos(x) - sqrt(41)*(5 + sqrt(41) + 5*cos(x) + sqrt(41)*cos(x) + (4*sin(x))/(1 + cos(x)) + 4*cos(x)*sin(x)/(1 + cos(x))) + 4*(5 + sqrt(41) + (4*sin(x))/(1 + cos(x)))*sin(x) + 5*sqrt(41)*cos(x) + 5*(4*sin(x))/(1 + cos(x)) + 20*cos(x)*sin(x)/(1 + cos(x)))/((5 + sqrt(41) + (4*sin(x))/(1 + cos(x)))*(5 + sqrt(41) + 5*cos(x) + sqrt(41)*cos(x) + (4*sin(x))/(1 + cos(x)) + 4*cos(x)*sin(x)/(1 + cos(x)))))/41
Combinatoria
[src]
/ ____ \
____ | 5 \/ 41 4*sin(x) |
\/ 41 *log|----------------------- - ----------------------- + --------------------------------------------------------------------|
| ____ 4*sin(x) ____ 4*sin(x) ____ ____ 4*sin(x) 4*cos(x)*sin(x)|
|5 + \/ 41 + ---------- 5 + \/ 41 + ---------- 5 + \/ 41 + 5*cos(x) + \/ 41 *cos(x) + ---------- + ---------------|
\ 1 + cos(x) 1 + cos(x) 1 + cos(x) 1 + cos(x) /
------------------------------------------------------------------------------------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{4 \sin{\left(x \right)}}{5 \cos{\left(x \right)} + \sqrt{41} \cos{\left(x \right)} + 5 + \sqrt{41} + \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} - \frac{\sqrt{41}}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} + \frac{5}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
sqrt(41)*log(5/(5 + sqrt(41) + 4*sin(x)/(1 + cos(x))) - sqrt(41)/(5 + sqrt(41) + 4*sin(x)/(1 + cos(x))) + 4*sin(x)/(5 + sqrt(41) + 5*cos(x) + sqrt(41)*cos(x) + 4*sin(x)/(1 + cos(x)) + 4*cos(x)*sin(x)/(1 + cos(x))))/41
Compilar la expresión
[src]
/ 4*sin(x) ____ \
|---------- - \/ 41 + 5|
____ |1 + cos(x) |
\/ 41 *log|-----------------------|
| ____ 4*sin(x) |
|5 + \/ 41 + ----------|
\ 1 + cos(x)/
-----------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{\left(- \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}\right) + 5}{\left(5 + \sqrt{41}\right) + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
sqrt(41)*log(((4*sin(x))/(1 + cos(x)) - sqrt(41) + 5)/(5 + sqrt(41) + (4*sin(x))/(1 + cos(x))))/41
Parte trigonométrica
[src]
/ ____ 4*sin(x) \
|5 - \/ 41 + ----------|
____ | 1 + cos(x)|
\/ 41 *log|-----------------------|
| ____ 4*sin(x) |
|5 + \/ 41 + ----------|
\ 1 + cos(x)/
-----------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
/ ____ 4 \
|5 - \/ 41 + ------------------------|
| / 1 \ |
| |1 + -----------|*csc(x)|
| | /pi \| |
| | csc|-- - x|| |
____ | \ \2 // |
\/ 41 *log|-------------------------------------|
| ____ 4 |
|5 + \/ 41 + ------------------------|
| / 1 \ |
| |1 + -----------|*csc(x)|
| | /pi \| |
| | csc|-- - x|| |
\ \ \2 // /
-------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}}{5 + \sqrt{41} + \frac{4}{\left(1 + \frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}\right) \csc{\left(x \right)}}} \right)}}{41}$$
/ ____ 4*sin(x) \
|5 - \/ 41 + ---------------|
| / pi\|
| 1 + sin|x + --||
____ | \ 2 /|
\/ 41 *log|----------------------------|
| ____ 4*sin(x) |
|5 + \/ 41 + ---------------|
| / pi\|
| 1 + sin|x + --||
\ \ 2 //
----------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}}{5 + \sqrt{41} + \frac{4 \sin{\left(x \right)}}{\sin{\left(x + \frac{\pi}{2} \right)} + 1}} \right)}}{41}$$
/ /x\ \
| 8*cot|-| |
| ____ \2/ |
|5 - \/ 41 + --------------------------------|
| / 2/x\\|
| | -1 + cot |-|||
| / 2/x\\ | \2/||
| |1 + cot |-||*|1 + ------------||
| \ \2// | 2/x\ ||
| | 1 + cot |-| ||
____ | \ \2/ /|
\/ 41 *log|---------------------------------------------|
| /x\ |
| 8*cot|-| |
| ____ \2/ |
|5 + \/ 41 + --------------------------------|
| / 2/x\\|
| | -1 + cot |-|||
| / 2/x\\ | \2/||
| |1 + cot |-||*|1 + ------------||
| \ \2// | 2/x\ ||
| | 1 + cot |-| ||
\ \ \2/ //
---------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{8 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{5 + \sqrt{41} + \frac{8 \cot{\left(\frac{x}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}} \right)}}{41}$$
/ ____ 4 \
|5 - \/ 41 + -------------------|
| / 1 \ |
| |1 + ------|*csc(x)|
____ | \ sec(x)/ |
\/ 41 *log|--------------------------------|
| ____ 4 |
|5 + \/ 41 + -------------------|
| / 1 \ |
| |1 + ------|*csc(x)|
\ \ sec(x)/ /
--------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}}{5 + \sqrt{41} + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \csc{\left(x \right)}}} \right)}}{41}$$
/ / pi\\
| 4*cos|x - --||
| ____ \ 2 /|
|5 - \/ 41 + -------------|
____ | 1 + cos(x) |
\/ 41 *log|--------------------------|
| / pi\|
| 4*cos|x - --||
| ____ \ 2 /|
|5 + \/ 41 + -------------|
\ 1 + cos(x) /
--------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}}{5 + \sqrt{41} + \frac{4 \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} + 1}} \right)}}{41}$$
/ /x\ \
| 8*tan|-| |
| ____ \2/ |
|5 - \/ 41 + -------------------------------|
| / 2/x\\|
| | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-||*|1 + -----------||
| \ \2// | 2/x\||
| | 1 + tan |-|||
____ | \ \2//|
\/ 41 *log|--------------------------------------------|
| /x\ |
| 8*tan|-| |
| ____ \2/ |
|5 + \/ 41 + -------------------------------|
| / 2/x\\|
| | 1 - tan |-|||
| / 2/x\\ | \2/||
| |1 + tan |-||*|1 + -----------||
| \ \2// | 2/x\||
| | 1 + tan |-|||
\ \ \2///
--------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{8 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}}{5 + \sqrt{41} + \frac{8 \tan{\left(\frac{x}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}} \right)}}{41}$$
/ ____ 4 \
|5 - \/ 41 + ------------------------|
| / 1 \ / pi\|
| |1 + ------|*sec|x - --||
____ | \ sec(x)/ \ 2 /|
\/ 41 *log|-------------------------------------|
| ____ 4 |
|5 + \/ 41 + ------------------------|
| / 1 \ / pi\|
| |1 + ------|*sec|x - --||
\ \ sec(x)/ \ 2 //
-------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} + 5 + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}}{5 + \sqrt{41} + \frac{4}{\left(1 + \frac{1}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}} \right)}}{41}$$
sqrt(41)*log((5 - sqrt(41) + 4/((1 + 1/sec(x))*sec(x - pi/2)))/(5 + sqrt(41) + 4/((1 + 1/sec(x))*sec(x - pi/2))))/41
Unión de expresiones racionales
[src]
/ ____ \
____ |5 + 4*sin(x) + 5*cos(x) - \/ 41 *(1 + cos(x))|
\/ 41 *log|---------------------------------------------|
| / ____\ |
\ 4*sin(x) + (1 + cos(x))*\5 + \/ 41 / /
---------------------------------------------------------
41
$$\frac{\sqrt{41} \log{\left(\frac{- \sqrt{41} \left(\cos{\left(x \right)} + 1\right) + 4 \sin{\left(x \right)} + 5 \cos{\left(x \right)} + 5}{\left(5 + \sqrt{41}\right) \left(\cos{\left(x \right)} + 1\right) + 4 \sin{\left(x \right)}} \right)}}{41}$$
sqrt(41)*log((5 + 4*sin(x) + 5*cos(x) - sqrt(41)*(1 + cos(x)))/(4*sin(x) + (1 + cos(x))*(5 + sqrt(41))))/41