Mister Exam
Expression simplification/
Least common denominator/
cos(x)/(2*sqrt(x))-sqrt(x)*sin(x)+6*x/asin(2*x)-2*(3*x^2+7)/(sqrt(1-4*x^2)*asin(2*x)^2)
Least common denominator cos(x)/(2*sqrt(x))-sqrt(x)*sin(x)+6*x/asin(2*x)-2*(3*x^2+7)/(sqrt(1-4*x^2)*asin(2*x)^2)
The solution
You have entered
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/ 2 \
cos(x) ___ 6*x 2*\3*x + 7/
------- - \/ x *sin(x) + --------- - ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$\left(\frac{6 x}{\operatorname{asin}{\left(2 x \right)}} + \left(- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}\right)\right) - \frac{2 \left(3 x^{2} + 7\right)}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}}$$
cos(x)/((2*sqrt(x))) - sqrt(x)*sin(x) + (6*x)/asin(2*x) - 2*(3*x^2 + 7)/(sqrt(1 - 4*x^2)*asin(2*x)^2)
General simplification
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2
cos(x) ___ 14 6*x 6*x
------- - \/ x *sin(x) - ------------------------ + --------- - ------------------------
___ __________ asin(2*x) __________
2*\/ x / 2 2 / 2 2
\/ 1 - 4*x *asin (2*x) \/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} - \frac{6 x^{2}}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
cos(x)/(2*sqrt(x)) - sqrt(x)*sin(x) - 14/(sqrt(1 - 4*x^2)*asin(2*x)^2) + 6*x/asin(2*x) - 6*x^2/(sqrt(1 - 4*x^2)*asin(2*x)^2)
Rational denominator
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__________ __________
3 3/2 2 7/2 2 3 3 3 2 3 5/2 / 2 ___ / 2
- asin (2*x)*cos(x) - 12*x *asin (2*x) + 48*x *asin (2*x) - 8*x *asin (2*x)*sin(x) + 2*x*asin (2*x)*sin(x) + 4*x *asin (2*x)*cos(x) + 12*x *\/ 1 - 4*x *asin(2*x) + 28*\/ x *\/ 1 - 4*x *asin(2*x)
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___ / 2\ 3
2*\/ x *\-1 + 4*x /*asin (2*x)
$$\frac{48 x^{\frac{7}{2}} \operatorname{asin}^{2}{\left(2 x \right)} + 12 x^{\frac{5}{2}} \sqrt{1 - 4 x^{2}} \operatorname{asin}{\left(2 x \right)} - 12 x^{\frac{3}{2}} \operatorname{asin}^{2}{\left(2 x \right)} + 28 \sqrt{x} \sqrt{1 - 4 x^{2}} \operatorname{asin}{\left(2 x \right)} - 8 x^{3} \sin{\left(x \right)} \operatorname{asin}^{3}{\left(2 x \right)} + 4 x^{2} \cos{\left(x \right)} \operatorname{asin}^{3}{\left(2 x \right)} + 2 x \sin{\left(x \right)} \operatorname{asin}^{3}{\left(2 x \right)} - \cos{\left(x \right)} \operatorname{asin}^{3}{\left(2 x \right)}}{2 \sqrt{x} \left(4 x^{2} - 1\right) \operatorname{asin}^{3}{\left(2 x \right)}}$$
(-asin(2*x)^3*cos(x) - 12*x^(3/2)*asin(2*x)^2 + 48*x^(7/2)*asin(2*x)^2 - 8*x^3*asin(2*x)^3*sin(x) + 2*x*asin(2*x)^3*sin(x) + 4*x^2*asin(2*x)^3*cos(x) + 12*x^(5/2)*sqrt(1 - 4*x^2)*asin(2*x) + 28*sqrt(x)*sqrt(1 - 4*x^2)*asin(2*x))/(2*sqrt(x)*(-1 + 4*x^2)*asin(2*x)^3)
Common denominator
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/ __________ __________ __________ \
| 5/2 ___ / 2 2 3/2 / 2 / 2 2 |
-\12*x + 28*\/ x - \/ 1 - 4*x *asin (2*x)*cos(x) - 12*x *\/ 1 - 4*x *asin(2*x) + 2*x*\/ 1 - 4*x *asin (2*x)*sin(x)/
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__________
___ / 2 2
2*\/ x *\/ 1 - 4*x *asin (2*x)
$$- \frac{12 x^{\frac{5}{2}} - 12 x^{\frac{3}{2}} \sqrt{1 - 4 x^{2}} \operatorname{asin}{\left(2 x \right)} + 28 \sqrt{x} + 2 x \sqrt{1 - 4 x^{2}} \sin{\left(x \right)} \operatorname{asin}^{2}{\left(2 x \right)} - \sqrt{1 - 4 x^{2}} \cos{\left(x \right)} \operatorname{asin}^{2}{\left(2 x \right)}}{2 \sqrt{x} \sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}}$$
-(12*x^(5/2) + 28*sqrt(x) - sqrt(1 - 4*x^2)*asin(2*x)^2*cos(x) - 12*x^(3/2)*sqrt(1 - 4*x^2)*asin(2*x) + 2*x*sqrt(1 - 4*x^2)*asin(2*x)^2*sin(x))/(2*sqrt(x)*sqrt(1 - 4*x^2)*asin(2*x)^2)
Numerical answer
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-x^0.5*sin(x) + 0.5*x^(-0.5)*cos(x) + 6.0*x/asin(2*x) - 0.5*(0.25 - x^2)^(-0.5)*(14.0 + 6.0*x^2)/asin(2*x)^2
-x^0.5*sin(x) + 0.5*x^(-0.5)*cos(x) + 6.0*x/asin(2*x) - 0.5*(0.25 - x^2)^(-0.5)*(14.0 + 6.0*x^2)/asin(2*x)^2
Combinatorics
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/ __________ __________ __________ \
| 5/2 ___ / 2 2 3/2 / 2 / 2 2 |
-\12*x + 28*\/ x - \/ 1 - 4*x *asin (2*x)*cos(x) - 12*x *\/ 1 - 4*x *asin(2*x) + 2*x*\/ 1 - 4*x *asin (2*x)*sin(x)/
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___ _______________________ 2
2*\/ x *\/ -(1 + 2*x)*(-1 + 2*x) *asin (2*x)
$$- \frac{12 x^{\frac{5}{2}} - 12 x^{\frac{3}{2}} \sqrt{1 - 4 x^{2}} \operatorname{asin}{\left(2 x \right)} + 28 \sqrt{x} + 2 x \sqrt{1 - 4 x^{2}} \sin{\left(x \right)} \operatorname{asin}^{2}{\left(2 x \right)} - \sqrt{1 - 4 x^{2}} \cos{\left(x \right)} \operatorname{asin}^{2}{\left(2 x \right)}}{2 \sqrt{x} \sqrt{- \left(2 x - 1\right) \left(2 x + 1\right)} \operatorname{asin}^{2}{\left(2 x \right)}}$$
-(12*x^(5/2) + 28*sqrt(x) - sqrt(1 - 4*x^2)*asin(2*x)^2*cos(x) - 12*x^(3/2)*sqrt(1 - 4*x^2)*asin(2*x) + 2*x*sqrt(1 - 4*x^2)*asin(2*x)^2*sin(x))/(2*sqrt(x)*sqrt(-(1 + 2*x)*(-1 + 2*x))*asin(2*x)^2)
Powers
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I*x -I*x
e e
---- + ----- ___ / -I*x I*x\ 2
2 2 6*x I*\/ x *\- e + e / 14 + 6*x
------------ + --------- + ------------------------ - ------------------------
___ asin(2*x) 2 __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$\frac{i \sqrt{x} \left(e^{i x} - e^{- i x}\right)}{2} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}}{2 \sqrt{x}}$$
2
cos(x) ___ 6*x -14 - 6*x
------- - \/ x *sin(x) + --------- + ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} + \frac{- 6 x^{2} - 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
2
cos(x) ___ 6*x 14 + 6*x
------- - \/ x *sin(x) + --------- - ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
cos(x)/(2*sqrt(x)) - sqrt(x)*sin(x) + 6*x/asin(2*x) - (14 + 6*x^2)/(sqrt(1 - 4*x^2)*asin(2*x)^2)
Assemble expression
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2
cos(x) ___ 6*x 14 + 6*x
------- - \/ x *sin(x) + --------- - ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
cos(x)/(2*sqrt(x)) - sqrt(x)*sin(x) + 6*x/asin(2*x) - (14 + 6*x^2)/(sqrt(1 - 4*x^2)*asin(2*x)^2)
Trigonometric part
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___ 2
1 \/ x 6*x 14 + 6*x
-------------- - ----------- + --------- - ------------------------
___ / pi\ asin(2*x) __________
2*\/ x *sec(x) sec|x - --| / 2 2
\ 2 / \/ 1 - 4*x *asin (2*x)
$$- \frac{\sqrt{x}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{1}{2 \sqrt{x} \sec{\left(x \right)}}$$
2/x\ ___ /x\
1 - tan |-| 2 2*\/ x *tan|-|
6*x \2/ 14 + 6*x \2/
--------- + --------------------- - ------------------------ - --------------
asin(2*x) ___ / 2/x\\ __________ 2/x\
2*\/ x *|1 + tan |-|| / 2 2 1 + tan |-|
\ \2// \/ 1 - 4*x *asin (2*x) \2/
$$- \frac{2 \sqrt{x} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{2 \sqrt{x} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
2
cos(x) ___ 6*x 14 + 6*x
------- - \/ x *sin(x) + --------- - ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
/ 2 \
cos(x) ___ 2*\3*x + 7/ 6*x
------- - \/ x *sin(x) - ------------------------ + ---------
___ __________ asin(2*x)
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{2 \left(3 x^{2} + 7\right)}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
2
cos(x) ___ / pi\ 6*x 14 + 6*x
------- - \/ x *cos|x - --| + --------- - ------------------------
___ \ 2 / asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \cos{\left(x - \frac{\pi}{2} \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}}$$
___ 2
1 \/ x 6*x 14 + 6*x
-------------- - ------ + --------- - ------------------------
___ csc(x) asin(2*x) __________
2*\/ x *sec(x) / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \frac{\sqrt{x}}{\csc{\left(x \right)}} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{1}{2 \sqrt{x} \sec{\left(x \right)}}$$
2/x\ ___ /x\
-1 + cot |-| 2 2*\/ x *cot|-|
6*x \2/ 14 + 6*x \2/
--------- + --------------------- - ------------------------ - --------------
asin(2*x) ___ / 2/x\\ __________ 2/x\
2*\/ x *|1 + cot |-|| / 2 2 1 + cot |-|
\ \2// \/ 1 - 4*x *asin (2*x) \2/
$$- \frac{2 \sqrt{x} \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{2 \sqrt{x} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ 2
1 \/ x 6*x 14 + 6*x
------------------- - ------ + --------- - ------------------------
___ /pi \ csc(x) asin(2*x) __________
2*\/ x *csc|-- - x| / 2 2
\2 / \/ 1 - 4*x *asin (2*x)
$$- \frac{\sqrt{x}}{\csc{\left(x \right)}} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{1}{2 \sqrt{x} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ pi\
sin|x + --| 2
\ 2 / ___ 6*x 14 + 6*x
----------- - \/ x *sin(x) + --------- - ------------------------
___ asin(2*x) __________
2*\/ x / 2 2
\/ 1 - 4*x *asin (2*x)
$$- \sqrt{x} \sin{\left(x \right)} + \frac{6 x}{\operatorname{asin}{\left(2 x \right)}} - \frac{6 x^{2} + 14}{\sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}} + \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{2 \sqrt{x}}$$
sin(x + pi/2)/(2*sqrt(x)) - sqrt(x)*sin(x) + 6*x/asin(2*x) - (14 + 6*x^2)/(sqrt(1 - 4*x^2)*asin(2*x)^2)
Combining rational expressions
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__________
___ / 2\ / 2 / 3/2 \
- 4*\/ x *\7 + 3*x / + \/ 1 - 4*x *\12*x + (-2*x*sin(x) + cos(x))*asin(2*x)/*asin(2*x)
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__________
___ / 2 2
2*\/ x *\/ 1 - 4*x *asin (2*x)
$$\frac{- 4 \sqrt{x} \left(3 x^{2} + 7\right) + \sqrt{1 - 4 x^{2}} \left(12 x^{\frac{3}{2}} + \left(- 2 x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \operatorname{asin}{\left(2 x \right)}\right) \operatorname{asin}{\left(2 x \right)}}{2 \sqrt{x} \sqrt{1 - 4 x^{2}} \operatorname{asin}^{2}{\left(2 x \right)}}$$
(-4*sqrt(x)*(7 + 3*x^2) + sqrt(1 - 4*x^2)*(12*x^(3/2) + (-2*x*sin(x) + cos(x))*asin(2*x))*asin(2*x))/(2*sqrt(x)*sqrt(1 - 4*x^2)*asin(2*x)^2)