Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/(2*sqrt(cos(x)))-sin(sqrt(x))/(2*sqrt(x))
¿Cómo vas a descomponer esta sin(x)/(2*sqrt(cos(x)))-sin(sqrt(x))/(2*sqrt(x)) expresión en fracciones?
Solución
Ha introducido
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/ ___\
sin(x) sin\\/ x /
------------ - ----------
________ ___
2*\/ cos(x) 2*\/ x
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)}}} - \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$$
sin(x)/((2*sqrt(cos(x)))) - sin(sqrt(x))/(2*sqrt(x))
Respuesta numérica
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0.5*cos(x)^(-0.5)*sin(x) - 0.5*x^(-0.5)*sin(sqrt(x))
0.5*cos(x)^(-0.5)*sin(x) - 0.5*x^(-0.5)*sin(sqrt(x))
Unión de expresiones racionales
[src]
___ ________ / ___\
\/ x *sin(x) - \/ cos(x) *sin\\/ x /
------------------------------------
___ ________
2*\/ x *\/ cos(x)
$$\frac{\sqrt{x} \sin{\left(x \right)} - \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}}$$
(sqrt(x)*sin(x) - sqrt(cos(x))*sin(sqrt(x)))/(2*sqrt(x)*sqrt(cos(x)))
Potencias
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/ ___ ___\
/ -I*x I*x\ | -I*\/ x I*\/ x |
I*\- e + e / I*\- e + e /
- --------------------- + --------------------------
______________ ___
/ I*x -I*x 4*\/ x
/ e e
4* / ---- + -----
\/ 2 2
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{4 \sqrt{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}} + \frac{i \left(e^{i \sqrt{x}} - e^{- i \sqrt{x}}\right)}{4 \sqrt{x}}$$
-i*(-exp(-i*x) + exp(i*x))/(4*sqrt(exp(i*x)/2 + exp(-i*x)/2)) + i*(-exp(-i*sqrt(x)) + exp(i*sqrt(x)))/(4*sqrt(x))
Denominador común
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/ ________ / ___\ ___ \
-\\/ cos(x) *sin\\/ x / - \/ x *sin(x)/
----------------------------------------
___ ________
2*\/ x *\/ cos(x)
$$- \frac{- \sqrt{x} \sin{\left(x \right)} + \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}}$$
-(sqrt(cos(x))*sin(sqrt(x)) - sqrt(x)*sin(x))/(2*sqrt(x)*sqrt(cos(x)))
Denominador racional
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___ / ___\ ________
- 2*\/ x *cos(x)*sin\\/ x / + 2*x*\/ cos(x) *sin(x)
---------------------------------------------------
4*x*cos(x)
$$\frac{- 2 \sqrt{x} \sin{\left(\sqrt{x} \right)} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} \sqrt{\cos{\left(x \right)}}}{4 x \cos{\left(x \right)}}$$
(-2*sqrt(x)*cos(x)*sin(sqrt(x)) + 2*x*sqrt(cos(x))*sin(x))/(4*x*cos(x))
Combinatoria
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/ ________ / ___\ ___ \
-\\/ cos(x) *sin\\/ x / - \/ x *sin(x)/
----------------------------------------
___ ________
2*\/ x *\/ cos(x)
$$- \frac{- \sqrt{x} \sin{\left(x \right)} + \sin{\left(\sqrt{x} \right)} \sqrt{\cos{\left(x \right)}}}{2 \sqrt{x} \sqrt{\cos{\left(x \right)}}}$$
-(sqrt(cos(x))*sin(sqrt(x)) - sqrt(x)*sin(x))/(2*sqrt(x)*sqrt(cos(x)))
Parte trigonométrica
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/ pi\ / ___ pi\
cos|x - --| cos|\/ x - --|
\ 2 / \ 2 /
------------ - ---------------
________ ___
2*\/ cos(x) 2*\/ x
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{\cos{\left(x \right)}}} - \frac{\cos{\left(\sqrt{x} - \frac{\pi}{2} \right)}}{2 \sqrt{x}}$$
1 1
-------------------------- - -----------------------
________ ___ / ___ pi\
/ 1 / pi\ 2*\/ x *sec|\/ x - --|
2* / ------ *sec|x - --| \ 2 /
\/ sec(x) \ 2 /
$$\frac{1}{2 \sqrt{\frac{1}{\sec{\left(x \right)}}} \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{2 \sqrt{x} \sec{\left(\sqrt{x} - \frac{\pi}{2} \right)}}$$
/ ___\
sin(x) sin\\/ x /
------------------- - ----------
_____________ ___
/ / pi\ 2*\/ x
2* / sin|x + --|
\/ \ 2 /
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\sin{\left(x + \frac{\pi}{2} \right)}}} - \frac{\sin{\left(\sqrt{x} \right)}}{2 \sqrt{x}}$$
/ ___\
/x\ |\/ x |
tan|-| tan|-----|
\2/ \ 2 /
----------------------------------- - -----------------------
_____________ / / ___\\
/ 2/x\ ___ | 2|\/ x ||
/ 1 - tan |-| \/ x *|1 + tan |-----||
/ \2/ / 2/x\\ \ \ 2 //
/ ----------- *|1 + tan |-||
/ 2/x\ \ \2//
/ 1 + tan |-|
\/ \2/
$$\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\tan{\left(\frac{\sqrt{x}}{2} \right)}}{\sqrt{x} \left(\tan^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)}$$
/ ___\
/x\ |\/ x |
cot|-| cot|-----|
\2/ \ 2 /
------------------------------------ - -----------------------
______________ / / ___\\
/ 2/x\ ___ | 2|\/ x ||
/ -1 + cot |-| \/ x *|1 + cot |-----||
/ \2/ / 2/x\\ \ \ 2 //
/ ------------ *|1 + cot |-||
/ 2/x\ \ \2//
/ 1 + cot |-|
\/ \2/
$$\frac{\cot{\left(\frac{x}{2} \right)}}{\sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\cot{\left(\frac{\sqrt{x}}{2} \right)}}{\sqrt{x} \left(\cot^{2}{\left(\frac{\sqrt{x}}{2} \right)} + 1\right)}$$
1 1
---------------------------- - ------------------
_____________ ___ / ___\
/ 1 2*\/ x *csc\\/ x /
2* / ----------- *csc(x)
/ /pi \
/ csc|-- - x|
\/ \2 /
$$\frac{1}{2 \sqrt{\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}}} \csc{\left(x \right)}} - \frac{1}{2 \sqrt{x} \csc{\left(\sqrt{x} \right)}}$$
1 1
--------------------- - ------------------
________ ___ / ___\
/ 1 2*\/ x *csc\\/ x /
2* / ------ *csc(x)
\/ sec(x)
$$\frac{1}{2 \sqrt{\frac{1}{\sec{\left(x \right)}}} \csc{\left(x \right)}} - \frac{1}{2 \sqrt{x} \csc{\left(\sqrt{x} \right)}}$$
1/(2*sqrt(1/sec(x))*csc(x)) - 1/(2*sqrt(x)*csc(sqrt(x)))