Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
acos(x)^(tan(x)^2)+x*acos(x)^(tan(x)^2)*((2+2*tan(x)^2)*log(acos(x))*tan(x)-tan(x)^2/(sqrt(1-x^2)*acos(x)))
¿Cómo vas a descomponer esta acos(x)^(tan(x)^2)+x*acos(x)^(tan(x)^2)*((2+2*tan(x)^2)*log(acos(x))*tan(x)-tan(x)^2/(sqrt(1-x^2)*acos(x))) expresión en fracciones?
Solución
Ha introducido
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2 2 / 2 \
tan (x) tan (x) |/ 2 \ tan (x) |
(acos(x)) + x*(acos(x)) *|\2 + 2*tan (x)/*log(acos(x))*tan(x) - -------------------|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x)/
$$x \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}\right) + \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
acos(x)^(tan(x)^2) + (x*acos(x)^(tan(x)^2))*(((2 + 2*tan(x)^2)*log(acos(x)))*tan(x) - tan(x)^2/(sqrt(1 - x^2)*acos(x)))
Simplificación general
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1
-1 + ------- 2 2
2 -1 + tan (x) 2 tan (x)
cos (x) x*(acos(x)) *tan (x) 2*x*(acos(x)) *log(acos(x))*tan(x)
(acos(x)) - ------------------------------- + ----------------------------------------
________ 2
/ 2 cos (x)
\/ 1 - x
$$\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{x \tan^{2}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)} - 1}{\left(x \right)}}{\sqrt{1 - x^{2}}} + \operatorname{acos}^{-1 + \frac{1}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
acos(x)^(-1 + cos(x)^(-2)) - x*acos(x)^(-1 + tan(x)^2)*tan(x)^2/sqrt(1 - x^2) + 2*x*acos(x)^(tan(x)^2)*log(acos(x))*tan(x)/cos(x)^2
Respuesta numérica
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acos(x)^(tan(x)^2) + x*acos(x)^(tan(x)^2)*((2.0 + 2.0*tan(x)^2)*log(acos(x))*tan(x) - (1.0 - x^2)^(-0.5)*tan(x)^2/acos(x))
acos(x)^(tan(x)^2) + x*acos(x)^(tan(x)^2)*((2.0 + 2.0*tan(x)^2)*log(acos(x))*tan(x) - (1.0 - x^2)^(-0.5)*tan(x)^2/acos(x))
Denominador común
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2
2 2 2 tan (x) 2
tan (x) tan (x) 3 tan (x) x*(acos(x)) *tan (x)
(acos(x)) + 2*x*(acos(x)) *tan (x)*log(acos(x)) + 2*x*(acos(x)) *log(acos(x))*tan(x) - --------------------------
________
/ 2
\/ 1 - x *acos(x)
$$2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + 2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} - \frac{x \tan^{2}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
acos(x)^(tan(x)^2) + 2*x*acos(x)^(tan(x)^2)*tan(x)^3*log(acos(x)) + 2*x*acos(x)^(tan(x)^2)*log(acos(x))*tan(x) - x*acos(x)^(tan(x)^2)*tan(x)^2/(sqrt(1 - x^2)*acos(x))
Potencias
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2 2 / 2 \
tan (x) tan (x) |/ 2 \ tan (x) |
(acos(x)) + x*(acos(x)) *|\2 + 2*tan (x)/*log(acos(x))*tan(x) - -------------------|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x)/
$$x \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2 2 / / 2\ \
/ I*x -I*x\ / I*x -I*x\ | | / I*x -I*x\ | |
-\- e + e / -\- e + e / | | 2*\- e + e / | / I*x -I*x\ |
------------------- ------------------- | I*|2 - -------------------|*\- e + e /*log(acos(x))|
2 2 | 2 | 2 | |
/ I*x -I*x\ / I*x -I*x\ | / I*x -I*x\ | / I*x -I*x\ | |
\e + e / \e + e / | \- e + e / \ \e + e / / |
(acos(x)) + x*(acos(x)) *|----------------------------------- + ---------------------------------------------------------|
| ________ 2 I*x -I*x |
| / 2 / I*x -I*x\ e + e |
\\/ 1 - x *\e + e / *acos(x) /
$$x \left(\frac{i \left(- \frac{2 \left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}} + 2\right) \left(- e^{i x} + e^{- i x}\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{e^{i x} + e^{- i x}} + \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\sqrt{1 - x^{2}} \left(e^{i x} + e^{- i x}\right)^{2} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}}}{\left(x \right)} + \operatorname{acos}^{- \frac{\left(- e^{i x} + e^{- i x}\right)^{2}}{\left(e^{i x} + e^{- i x}\right)^{2}}}{\left(x \right)}$$
acos(x)^(-(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2) + x*acos(x)^(-(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*((-exp(i*x) + exp(-i*x))^2/(sqrt(1 - x^2)*(exp(i*x) + exp(-i*x))^2*acos(x)) + i*(2 - 2*(-exp(i*x) + exp(-i*x))^2/(exp(i*x) + exp(-i*x))^2)*(-exp(i*x) + exp(-i*x))*log(acos(x))/(exp(i*x) + exp(-i*x)))
Combinatoria
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2 / ________ ________ ________ \
tan (x) | / 2 2 / 2 3 / 2 |
(acos(x)) *\\/ 1 - x *acos(x) - x*tan (x) + 2*x*\/ 1 - x *tan (x)*acos(x)*log(acos(x)) + 2*x*\/ 1 - x *acos(x)*log(acos(x))*tan(x)/
-----------------------------------------------------------------------------------------------------------------------------------------------
___________________
\/ -(1 + x)*(-1 + x) *acos(x)
$$\frac{\left(2 x \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} \operatorname{acos}{\left(x \right)} + 2 x \sqrt{1 - x^{2}} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} \operatorname{acos}{\left(x \right)} - x \tan^{2}{\left(x \right)} + \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \operatorname{acos}{\left(x \right)}}$$
acos(x)^(tan(x)^2)*(sqrt(1 - x^2)*acos(x) - x*tan(x)^2 + 2*x*sqrt(1 - x^2)*tan(x)^3*acos(x)*log(acos(x)) + 2*x*sqrt(1 - x^2)*acos(x)*log(acos(x))*tan(x))/(sqrt(-(1 + x)*(-1 + x))*acos(x))
Denominador racional
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2 2 ________ 2 2 2 2 2
tan (x) 2 tan (x) / 2 tan (x) 2 tan (x) 3 tan (x) 3 tan (x) 3 3 tan (x)
- (acos(x)) *acos(x) + x *(acos(x)) *acos(x) + x*\/ 1 - x *(acos(x)) *tan (x) - 2*x*(acos(x)) *tan (x)*acos(x)*log(acos(x)) - 2*x*(acos(x)) *acos(x)*log(acos(x))*tan(x) + 2*x *(acos(x)) *tan (x)*acos(x)*log(acos(x)) + 2*x *(acos(x)) *acos(x)*log(acos(x))*tan(x)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2\
\-1 + x /*acos(x)
$$\frac{2 x^{3} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + 2 x^{3} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + x^{2} \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + x \sqrt{1 - x^{2}} \tan^{2}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} - 2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} - 2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} - \operatorname{acos}{\left(x \right)} \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}}{\left(x^{2} - 1\right) \operatorname{acos}{\left(x \right)}}$$
(-acos(x)^(tan(x)^2)*acos(x) + x^2*acos(x)^(tan(x)^2)*acos(x) + x*sqrt(1 - x^2)*acos(x)^(tan(x)^2)*tan(x)^2 - 2*x*acos(x)^(tan(x)^2)*tan(x)^3*acos(x)*log(acos(x)) - 2*x*acos(x)^(tan(x)^2)*acos(x)*log(acos(x))*tan(x) + 2*x^3*acos(x)^(tan(x)^2)*tan(x)^3*acos(x)*log(acos(x)) + 2*x^3*acos(x)^(tan(x)^2)*acos(x)*log(acos(x))*tan(x))/((-1 + x^2)*acos(x))
Compilar la expresión
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2 2 / 2 \
tan (x) tan (x) |/ 2 \ tan (x) |
(acos(x)) + x*(acos(x)) *|\2 + 2*tan (x)/*log(acos(x))*tan(x) - -------------------|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x)/
$$x \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
acos(x)^(tan(x)^2) + x*acos(x)^(tan(x)^2)*((2 + 2*tan(x)^2)*log(acos(x))*tan(x) - tan(x)^2/(sqrt(1 - x^2)*acos(x)))
Parte trigonométrica
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2 2 // 2 \ \
sec (x) sec (x) || 2*sec (x) | |
------------ ------------ ||2 + ------------|*log(acos(x))*sec(x) |
2/ pi\ 2/ pi\ || 2/ pi\| |
sec |x - --| sec |x - --| || sec |x - --|| 2 |
\ 2 / \ 2 / |\ \ 2 // sec (x) |
(acos(x)) + x*(acos(x)) *|-------------------------------------- - --------------------------------|
| / pi\ ________ |
| sec|x - --| / 2 2/ pi\|
| \ 2 / \/ 1 - x *acos(x)*sec |x - --||
\ \ 2 //
$$x \left(\frac{\left(\frac{2 \sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{\sec^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}}\right) \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}}{\left(x \right)}$$
4
4*sin (x)
---------
2 / 4 \
sin (2*x) | 2*x*log(acos(x))*sin(x) 4*x*sin (x) |
(acos(x)) *|1 + ----------------------- - -----------------------------|
| 3/ pi\ ________ |
| sin |x + --| / 2 2 |
\ \ 2 / \/ 1 - x *acos(x)*sin (2*x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin{\left(x \right)}}{\sin^{3}{\left(x + \frac{\pi}{2} \right)}} - \frac{4 x \sin^{4}{\left(x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(2 x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}}}{\left(x \right)}$$
4 4 / / 4 \ \
4*sin (x) 4*sin (x) | 2 | 8*sin (x)| |
--------- --------- | 2*sin (x)*|2 + ---------|*log(acos(x))|
2 2 | 4 | 2 | |
sin (2*x) sin (2*x) | 4*sin (x) \ sin (2*x)/ |
(acos(x)) + x*(acos(x)) *|- ----------------------------- + --------------------------------------|
| ________ sin(2*x) |
| / 2 2 |
\ \/ 1 - x *acos(x)*sin (2*x) /
$$x \left(\frac{2 \left(\frac{8 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - \frac{4 \sin^{4}{\left(x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(2 x \right)} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}}}{\left(x \right)}$$
2/ pi\
cos |x - --|
\ 2 /
------------ / 2/ pi\ 2/ pi\ \
2 | x*cos |x - --| 4*x*cos |x - --|*log(acos(x))|
cos (x) | \ 2 / \ 2 / |
(acos(x)) *|1 - --------------------------- + -----------------------------|
| ________ 2 / pi\ |
| / 2 2 cos (x)*cos|2*x - --| |
\ \/ 1 - x *acos(x)*cos (x) \ 2 / /
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}} - \frac{x \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \cos^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
/ 2 \
| / 2/x\\ /x\|
2 | 2 4*x*|1 + tan |-|| *log(acos(x))*tan|-||
tan (x) | x*tan (x) \ \2// \2/|
(acos(x)) *|1 - ------------------- + --------------------------------------|
| ________ 3 |
| / 2 / 2/x\\ |
| \/ 1 - x *acos(x) |1 - tan |-|| |
\ \ \2// /
$$\left(\frac{4 x \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2 / 2 2 \
tan (x) | x*tan (x) 4*x*tan (x)*log(acos(x))|
(acos(x)) *|1 - ------------------- + ------------------------|
| ________ sin(2*x) |
| / 2 |
\ \/ 1 - x *acos(x) /
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2/pi \
csc |-- - x|
\2 /
------------ / 3/pi \ 2/pi \ \
2 | 2*x*csc |-- - x|*log(acos(x)) x*csc |-- - x| |
csc (x) | \2 / \2 / |
(acos(x)) *|1 + ----------------------------- - ---------------------------|
| csc(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*csc (x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{x \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc^{2}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)}$$
1 / 2 \
------- | / 2/x\\ /x\ |
2 | 4*x*|1 + cot |-|| *cot|-|*log(acos(x))|
cot (x) | x \ \2// \2/ |
(acos(x)) *|1 - --------------------------- + --------------------------------------|
| ________ 3 |
| / 2 2 / 2/x\\ |
| \/ 1 - x *acos(x)*cot (x) |-1 + cot |-|| |
\ \ \2// /
$$\left(\frac{4 x \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2} \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3}} - \frac{x}{\sqrt{1 - x^{2}} \cot^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{1}{\cot^{2}{\left(x \right)}}}{\left(x \right)}$$
/ /x\\
2 | 2 4*x*log(acos(x))*tan|-||
tan (x) | x*tan (x) \2/|
(acos(x)) *|1 - ------------------- + -----------------------|
| ________ 3 |
| / 2 / 2/x\\ 4/x\|
| \/ 1 - x *acos(x) |1 - tan |-|| *cos |-||
\ \ \2// \2//
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(\frac{x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{3} \cos^{4}{\left(\frac{x}{2} \right)}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2/ pi\ 2/ pi\ // 2/ pi\\ \
cos |x - --| cos |x - --| || 2*cos |x - --|| |
\ 2 / \ 2 / || \ 2 /| / pi\ |
------------ ------------ ||2 + --------------|*cos|x - --|*log(acos(x)) 2/ pi\ |
2 2 || 2 | \ 2 / cos |x - --| |
cos (x) cos (x) |\ cos (x) / \ 2 / |
(acos(x)) + x*(acos(x)) *|--------------------------------------------- - ---------------------------|
| cos(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*cos (x)/
$$x \left(\frac{\left(2 + \frac{2 \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \cos^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
2
sec (x)
------------
2/ pi\
sec |x - --| / 3 2 \
\ 2 / | 2*x*sec (x)*log(acos(x)) 2*x*log(acos(x))*sec(x) x*sec (x) |
(acos(x)) *|1 + ------------------------ + ----------------------- - --------------------------------|
| 3/ pi\ / pi\ ________ |
| sec |x - --| sec|x - --| / 2 2/ pi\|
| \ 2 / \ 2 / \/ 1 - x *acos(x)*sec |x - --||
\ \ 2 //
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec^{3}{\left(x \right)}}{\sec^{3}{\left(x - \frac{\pi}{2} \right)}} + \frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{x \sec^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}}{\left(x \right)}$$
2/ pi\
cos |x - --|
\ 2 /
------------ / / pi\ 3/ pi\ 2/ pi\ \
2 | 2*x*cos|x - --|*log(acos(x)) 2*x*cos |x - --|*log(acos(x)) x*cos |x - --| |
cos (x) | \ 2 / \ 2 / \ 2 / |
(acos(x)) *|1 + ---------------------------- + ----------------------------- - ---------------------------|
| cos(x) 3 ________ |
| cos (x) / 2 2 |
\ \/ 1 - x *acos(x)*cos (x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}} + \frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cos^{3}{\left(x - \frac{\pi}{2} \right)}}{\cos^{3}{\left(x \right)}} - \frac{x \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \cos^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
2 / 2 \
tan (x) | 3 x*tan (x) |
(acos(x)) *|1 + 2*x*tan (x)*log(acos(x)) + 2*x*log(acos(x))*tan(x) - -------------------|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x)/
$$\left(2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan^{3}{\left(x \right)} + 2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2
sec (x)
------------
2/ pi\ / 2 / pi\\
sec |x - --| | 2 4*x*sec (x)*log(acos(x))*sec|2*x - --||
\ 2 / | x*sec (x) \ 2 /|
(acos(x)) *|1 - -------------------------------- + --------------------------------------|
| ________ 2/ pi\ |
| / 2 2/ pi\ sec |x - --| |
| \/ 1 - x *acos(x)*sec |x - --| \ 2 / |
\ \ 2 / /
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec^{2}{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} - \frac{x \sec^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}}{\left(x \right)}$$
2
sec (x)
------------
2/ pi\
sec |x - --| / 3 2 \
\ 2 / | 2*x*sec (x)*log(acos(x)) x*sec (x) |
(acos(x)) *|1 + ------------------------ - --------------------------------|
| / pi\ ________ |
| sec|x - --| / 2 2/ pi\|
| \ 2 / \/ 1 - x *acos(x)*sec |x - --||
\ \ 2 //
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec^{3}{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} - \frac{x \sec^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}}{\left(x \right)}$$
/ /x\ \
2 | 2 4*x*cot|-|*log(acos(x))|
tan (x) | x*tan (x) \2/ |
(acos(x)) *|1 - ------------------- + -----------------------|
| ________ 3 |
| / 2 / 2/x\\ 4/x\|
| \/ 1 - x *acos(x) |-1 + cot |-|| *sin |-||
\ \ \2// \2//
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{3} \sin^{4}{\left(\frac{x}{2} \right)}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
1
-------
2
cot (x) / 2*x*log(acos(x)) 2*x*log(acos(x)) x \
(acos(x)) *|1 + ---------------- + ---------------- - ---------------------------|
| cot(x) 3 ________ |
| cot (x) / 2 2 |
\ \/ 1 - x *acos(x)*cot (x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{\cot{\left(x \right)}} + \frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{\cot^{3}{\left(x \right)}} - \frac{x}{\sqrt{1 - x^{2}} \cot^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{1}{\cot^{2}{\left(x \right)}}}{\left(x \right)}$$
2/pi \ 2/pi \ // 2/pi \\ \
csc |-- - x| csc |-- - x| || 2*csc |-- - x|| |
\2 / \2 / || \2 /| /pi \ |
------------ ------------ ||2 + --------------|*csc|-- - x|*log(acos(x)) 2/pi \ |
2 2 || 2 | \2 / csc |-- - x| |
csc (x) csc (x) |\ csc (x) / \2 / |
(acos(x)) + x*(acos(x)) *|--------------------------------------------- - ---------------------------|
| csc(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*csc (x)/
$$x \left(\frac{\left(2 + \frac{2 \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc^{2}{\left(x \right)}}\right) \operatorname{acos}^{\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)}$$
2 / 2 \
tan (x) | x*tan (x) 2*x*log(acos(x))*sin(x)|
(acos(x)) *|1 - ------------------- + -----------------------|
| ________ 3 |
| / 2 cos (x) |
\ \/ 1 - x *acos(x) /
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
4
4*sin (x)
---------
2 / 4 4 \
sin (2*x) | 16*x*sin (x)*log(acos(x)) 4*x*sin (x) |
(acos(x)) *|1 + ------------------------- - -----------------------------|
| 3 ________ |
| sin (2*x) / 2 2 |
\ \/ 1 - x *acos(x)*sin (2*x)/
$$\left(\frac{16 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin^{4}{\left(x \right)}}{\sin^{3}{\left(2 x \right)}} - \frac{4 x \sin^{4}{\left(x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(2 x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}}}{\left(x \right)}$$
2 2 / 2 \
tan (x) tan (x) |/ 2 \ tan (x) |
(acos(x)) + x*(acos(x)) *|\2 + 2*tan (x)/*log(acos(x))*tan(x) - -------------------|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x)/
$$x \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)} + \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
4
4*sin (x)
---------
2 / 2 6 4 \
sin (2*x) | 4*x*sin (x)*log(acos(x)) 16*x*sin (x)*log(acos(x)) 4*x*sin (x) |
(acos(x)) *|1 + ------------------------ + ------------------------- - -----------------------------|
| sin(2*x) 3 ________ |
| sin (2*x) / 2 2 |
\ \/ 1 - x *acos(x)*sin (2*x)/
$$\left(\frac{16 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin^{6}{\left(x \right)}}{\sin^{3}{\left(2 x \right)}} + \frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} - \frac{4 x \sin^{4}{\left(x \right)}}{\sqrt{1 - x^{2}} \sin^{2}{\left(2 x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}}}{\left(x \right)}$$
2/pi \
csc |-- - x|
\2 /
------------ / /pi \ 3/pi \ 2/pi \ \
2 | 2*x*csc|-- - x|*log(acos(x)) 2*x*csc |-- - x|*log(acos(x)) x*csc |-- - x| |
csc (x) | \2 / \2 / \2 / |
(acos(x)) *|1 + ---------------------------- + ----------------------------- - ---------------------------|
| csc(x) 3 ________ |
| csc (x) / 2 2 |
\ \/ 1 - x *acos(x)*csc (x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}} + \frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \csc^{3}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{3}{\left(x \right)}} - \frac{x \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc^{2}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)}$$
2/ pi\
cos |x - --|
\ 2 /
------------ / / pi\ 2/ pi\ \
2 | 2*x*cos|x - --|*log(acos(x)) x*cos |x - --| |
cos (x) | \ 2 / \ 2 / |
(acos(x)) *|1 + ---------------------------- - ---------------------------|
| 3 ________ |
| cos (x) / 2 2 |
\ \/ 1 - x *acos(x)*cos (x)/
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \cos{\left(x - \frac{\pi}{2} \right)}}{\cos^{3}{\left(x \right)}} - \frac{x \cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \cos^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
1
-------
2 / / 2 \ \
cot (x) | x 2*x*\1 + cot (x)/*log(acos(x))|
(acos(x)) *|1 - --------------------------- + ------------------------------|
| ________ 3 |
| / 2 2 cot (x) |
\ \/ 1 - x *acos(x)*cot (x) /
$$\left(\frac{2 x \left(\cot^{2}{\left(x \right)} + 1\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{\cot^{3}{\left(x \right)}} - \frac{x}{\sqrt{1 - x^{2}} \cot^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{1}{\cot^{2}{\left(x \right)}}}{\left(x \right)}$$
2 2 // 2 \ \
sin (x) sin (x) || 2*sin (x)| |
------- ------- ||2 + ---------|*log(acos(x))*sin(x) |
2 2 || 2 | 2 |
cos (x) cos (x) |\ cos (x) / sin (x) |
(acos(x)) + x*(acos(x)) *|----------------------------------- - ---------------------------|
| cos(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*cos (x)/
$$x \left(\frac{\left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \cos^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}}{\left(x \right)}$$
2/pi \
csc |-- - x|
\2 /
------------ / 2/pi \ 2/pi \ \
2 | x*csc |-- - x| 4*x*csc |-- - x|*csc(2*x)*log(acos(x))|
csc (x) | \2 / \2 / |
(acos(x)) *|1 - --------------------------- + --------------------------------------|
| ________ 2 |
| / 2 2 csc (x) |
\ \/ 1 - x *acos(x)*csc (x) /
$$\left(\frac{4 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \csc{\left(2 x \right)} \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}} - \frac{x \csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc^{2}{\left(x \right)}} + 1\right) \operatorname{acos}^{\frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)}$$
2 / 2 \
tan (x) | x*tan (x) 2*x*log(acos(x))*tan(x)|
(acos(x)) *|1 - ------------------- + -----------------------|
| ________ 2 |
| / 2 cos (x) |
\ \/ 1 - x *acos(x) /
$$\left(\frac{2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
1 1 // 2 \ \
------- ------- ||2 + -------|*log(acos(x)) |
2 2 || 2 | |
cot (x) cot (x) |\ cot (x)/ 1 |
(acos(x)) + x*(acos(x)) *|-------------------------- - ---------------------------|
| cot(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*cot (x)/
$$x \left(\frac{\left(2 + \frac{2}{\cot^{2}{\left(x \right)}}\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)}}{\cot{\left(x \right)}} - \frac{1}{\sqrt{1 - x^{2}} \cot^{2}{\left(x \right)} \operatorname{acos}{\left(x \right)}}\right) \operatorname{acos}^{\frac{1}{\cot^{2}{\left(x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{1}{\cot^{2}{\left(x \right)}}}{\left(x \right)}$$
2 / 2 \
tan (x) | x*tan (x) / 2 \ |
(acos(x)) *|1 - ------------------- + 2*x*\1 + tan (x)/*log(acos(x))*tan(x)|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x) /
$$\left(2 x \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \tan{\left(x \right)} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
2 2 // 2 \ \
sec (x) sec (x) || 2*sec (x)| |
------- ------- ||2 + ---------|*log(acos(x))*sec(x) |
2 2 || 2 | 2 |
csc (x) csc (x) |\ csc (x) / sec (x) |
(acos(x)) + x*(acos(x)) *|----------------------------------- - ---------------------------|
| csc(x) ________ |
| / 2 2 |
\ \/ 1 - x *acos(x)*csc (x)/
$$x \left(\frac{\left(2 + \frac{2 \sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sec{\left(x \right)}}{\csc{\left(x \right)}} - \frac{\sec^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)} \csc^{2}{\left(x \right)}}\right) \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)} + \operatorname{acos}^{\frac{\sec^{2}{\left(x \right)}}{\csc^{2}{\left(x \right)}}}{\left(x \right)}$$
2 / 2 \
tan (x) | x*tan (x) 3 |
(acos(x)) *|1 - ------------------- + 2*x*sec (x)*log(acos(x))*sin(x)|
| ________ |
| / 2 |
\ \/ 1 - x *acos(x) /
$$\left(2 x \log{\left(\operatorname{acos}{\left(x \right)} \right)} \sin{\left(x \right)} \sec^{3}{\left(x \right)} - \frac{x \tan^{2}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}} + 1\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}$$
acos(x)^(tan(x)^2)*(1 - x*tan(x)^2/(sqrt(1 - x^2)*acos(x)) + 2*x*sec(x)^3*log(acos(x))*sin(x))
Unión de expresiones racionales
[src]
2 / ________ / ________ \ \
tan (x) | / 2 | / 2 / 2 \ | |
(acos(x)) *\\/ 1 - x *acos(x) + x*\-tan(x) + 2*\/ 1 - x *\1 + tan (x)/*acos(x)*log(acos(x))/*tan(x)/
--------------------------------------------------------------------------------------------------------------
________
/ 2
\/ 1 - x *acos(x)
$$\frac{\left(x \left(2 \sqrt{1 - x^{2}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\operatorname{acos}{\left(x \right)} \right)} \operatorname{acos}{\left(x \right)} - \tan{\left(x \right)}\right) \tan{\left(x \right)} + \sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}\right) \operatorname{acos}^{\tan^{2}{\left(x \right)}}{\left(x \right)}}{\sqrt{1 - x^{2}} \operatorname{acos}{\left(x \right)}}$$
acos(x)^(tan(x)^2)*(sqrt(1 - x^2)*acos(x) + x*(-tan(x) + 2*sqrt(1 - x^2)*(1 + tan(x)^2)*acos(x)*log(acos(x)))*tan(x))/(sqrt(1 - x^2)*acos(x))