Mister Exam
Least common denominator (x-sin(2*x)/2)/2-2*cos(x)+x
The solution
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sin(2*x)
x - --------
2
------------ - 2*cos(x) + x
2
$$x + \left(\frac{x - \frac{\sin{\left(2 x \right)}}{2}}{2} - 2 \cos{\left(x \right)}\right)$$
(x - sin(2*x)/2)/2 - 2*cos(x) + x
General simplification
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sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
-2*cos(x) - sin(2*x)/4 + 3*x/2
Rational denominator
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-sin(2*x) - 8*cos(x) + 6*x
--------------------------
4
$$\frac{6 x - \sin{\left(2 x \right)} - 8 \cos{\left(x \right)}}{4}$$
(-sin(2*x) - 8*cos(x) + 6*x)/4
Powers
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/ -2*I*x 2*I*x\
I*x -I*x 3*x I*\- e + e /
- e - e + --- + ----------------------
2 8
$$\frac{3 x}{2} + \frac{i \left(e^{2 i x} - e^{- 2 i x}\right)}{8} - e^{i x} - e^{- i x}$$
sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
-2*cos(x) - sin(2*x)/4 + 3*x/2
Combining rational expressions
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-sin(2*x) - 8*cos(x) + 6*x
--------------------------
4
$$\frac{6 x - \sin{\left(2 x \right)} - 8 \cos{\left(x \right)}}{4}$$
(-sin(2*x) - 8*cos(x) + 6*x)/4
Combinatorics
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sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
-2*cos(x) - sin(2*x)/4 + 3*x/2
Common denominator
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sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
-2*cos(x) - sin(2*x)/4 + 3*x/2
Assemble expression
[src]
sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
-2*cos(x) - sin(2*x)/4 + 3*x/2
Trigonometric part
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2 1 3*x
- ----------- - ---------- + ---
/pi \ 4*csc(2*x) 2
csc|-- - x|
\2 /
$$\frac{3 x}{2} - \frac{2}{\csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{4 \csc{\left(2 x \right)}}$$
/ pi\
cos|2*x - --|
\ 2 / 3*x
-2*cos(x) - ------------- + ---
4 2
$$\frac{3 x}{2} - 2 \cos{\left(x \right)} - \frac{\cos{\left(2 x - \frac{\pi}{2} \right)}}{4}$$
2 1 3*x - ------ - ---------- + --- sec(x) 4*csc(2*x) 2
$$\frac{3 x}{2} - \frac{2}{\sec{\left(x \right)}} - \frac{1}{4 \csc{\left(2 x \right)}}$$
/ 2/x\\
2*|-1 + cot |-||
3*x \ \2// cot(x)
--- - ---------------- - ---------------
2 2/x\ / 2 \
1 + cot |-| 2*\1 + cot (x)/
\2/
$$\frac{3 x}{2} - \frac{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\cot{\left(x \right)}}{2 \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/ pi\ sin(2*x) 3*x
- 2*sin|x + --| - -------- + ---
\ 2 / 4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \sin{\left(x + \frac{\pi}{2} \right)}$$
/ 2/x\\
2*|1 - tan |-||
3*x \ \2// tan(x)
--- - --------------- - ---------------
2 2/x\ / 2 \
1 + tan |-| 2*\1 + tan (x)/
\2/
$$\frac{3 x}{2} - \frac{2 \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} - \frac{\tan{\left(x \right)}}{2 \left(\tan^{2}{\left(x \right)} + 1\right)}$$
sin(2*x) 3*x
-2*cos(x) - -------- + ---
4 2
$$\frac{3 x}{2} - \frac{\sin{\left(2 x \right)}}{4} - 2 \cos{\left(x \right)}$$
2 1 3*x
- ------ - --------------- + ---
sec(x) / pi\ 2
4*sec|2*x - --|
\ 2 /
$$\frac{3 x}{2} - \frac{1}{4 \sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{2}{\sec{\left(x \right)}}$$
-2/sec(x) - 1/(4*sec(2*x - pi/2)) + 3*x/2