Mister Exam
Least common denominator sin(3*x)/6+(2*x*sin(2*x)+cos(2*x))/4+sin(x)/2
The solution
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sin(3*x) 2*x*sin(2*x) + cos(2*x) sin(x) -------- + ----------------------- + ------ 6 4 2
$$\left(\frac{2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}}{4} + \frac{\sin{\left(3 x \right)}}{6}\right) + \frac{\sin{\left(x \right)}}{2}$$
sin(3*x)/6 + ((2*x)*sin(2*x) + cos(2*x))/4 + sin(x)/2
General simplification
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sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2
Numerical answer
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0.25*cos(2*x) + 0.5*sin(x) + 0.166666666666667*sin(3*x) + 0.5*x*sin(2*x)
0.25*cos(2*x) + 0.5*sin(x) + 0.166666666666667*sin(3*x) + 0.5*x*sin(2*x)
Rational denominator
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8*sin(3*x) + 12*cos(2*x) + 24*sin(x) + 24*x*sin(2*x)
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48
$$\frac{24 x \sin{\left(2 x \right)} + 24 \sin{\left(x \right)} + 8 \sin{\left(3 x \right)} + 12 \cos{\left(2 x \right)}}{48}$$
(8*sin(3*x) + 12*cos(2*x) + 24*sin(x) + 24*x*sin(2*x))/48
Powers
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-2*I*x 2*I*x / -I*x I*x\ / -3*I*x 3*I*x\ / -2*I*x 2*I*x\ e e I*\- e + e / I*\- e + e / I*x*\- e + e / ------- + ------ - ------------------ - ---------------------- - ------------------------ 8 8 4 12 4
$$- \frac{i x \left(e^{2 i x} - e^{- 2 i x}\right)}{4} - \frac{i \left(e^{i x} - e^{- i x}\right)}{4} - \frac{i \left(e^{3 i x} - e^{- 3 i x}\right)}{12} + \frac{e^{2 i x}}{8} + \frac{e^{- 2 i x}}{8}$$
sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2
Combining rational expressions
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2*sin(3*x) + 3*cos(2*x) + 6*sin(x) + 6*x*sin(2*x)
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12
$$\frac{6 x \sin{\left(2 x \right)} + 6 \sin{\left(x \right)} + 2 \sin{\left(3 x \right)} + 3 \cos{\left(2 x \right)}}{12}$$
(2*sin(3*x) + 3*cos(2*x) + 6*sin(x) + 6*x*sin(2*x))/12
Common denominator
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sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2
Combinatorics
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sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2
Assemble expression
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sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2
Trigonometric part
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1 1 1 x
-------- + --------------- + ---------- + ----------
2*csc(x) /pi \ 6*csc(3*x) 2*csc(2*x)
4*csc|-- - 2*x|
\2 /
$$\frac{x}{2 \csc{\left(2 x \right)}} + \frac{1}{4 \csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{1}{6 \csc{\left(3 x \right)}} + \frac{1}{2 \csc{\left(x \right)}}$$
/x\ /3*x\
cot|-| cot|---| 2
\2/ \ 2 / -1 + cot (x) x*cot(x)
----------- + ----------------- + --------------- + -----------
2/x\ / 2/3*x\\ / 2 \ 2
1 + cot |-| 3*|1 + cot |---|| 4*\1 + cot (x)/ 1 + cot (x)
\2/ \ \ 2 //
$$\frac{x \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} + \frac{\cot^{2}{\left(x \right)} - 1}{4 \left(\cot^{2}{\left(x \right)} + 1\right)} + \frac{\cot{\left(\frac{3 x}{2} \right)}}{3 \left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right)} + \frac{\cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
/x\ /3*x\
tan|-| tan|---| 2
\2/ \ 2 / 1 - tan (x) x*tan(x)
----------- + ----------------- + --------------- + -----------
2/x\ / 2/3*x\\ / 2 \ 2
1 + tan |-| 3*|1 + tan |---|| 4*\1 + tan (x)/ 1 + tan (x)
\2/ \ \ 2 //
$$\frac{x \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + \frac{1 - \tan^{2}{\left(x \right)}}{4 \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{\tan{\left(\frac{3 x}{2} \right)}}{3 \left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right)} + \frac{\tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
1 1 1 x -------- + ---------- + ---------- + ---------- 2*csc(x) 4*sec(2*x) 6*csc(3*x) 2*csc(2*x)
$$\frac{x}{2 \csc{\left(2 x \right)}} + \frac{1}{4 \sec{\left(2 x \right)}} + \frac{1}{6 \csc{\left(3 x \right)}} + \frac{1}{2 \csc{\left(x \right)}}$$
/ pi\ / pi\ / pi\
cos|x - --| cos|3*x - --| x*cos|2*x - --|
\ 2 / cos(2*x) \ 2 / \ 2 /
----------- + -------- + ------------- + ---------------
2 4 6 2
$$\frac{x \cos{\left(2 x - \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{\cos{\left(3 x - \frac{\pi}{2} \right)}}{6}$$
1 1 1 x
------------- + ---------- + --------------- + ---------------
/ pi\ 4*sec(2*x) / pi\ / pi\
2*sec|x - --| 6*sec|3*x - --| 2*sec|2*x - --|
\ 2 / \ 2 / \ 2 /
$$\frac{x}{2 \sec{\left(2 x - \frac{\pi}{2} \right)}} + \frac{1}{6 \sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{1}{4 \sec{\left(2 x \right)}}$$
/pi \
sin|-- + 2*x|
sin(x) \2 / sin(3*x) x*sin(2*x)
------ + ------------- + -------- + ----------
2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\sin{\left(2 x + \frac{\pi}{2} \right)}}{4}$$
sin(x) cos(2*x) sin(3*x) x*sin(2*x) ------ + -------- + -------- + ---------- 2 4 6 2
$$\frac{x \sin{\left(2 x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(3 x \right)}}{6} + \frac{\cos{\left(2 x \right)}}{4}$$
sin(x)/2 + cos(2*x)/4 + sin(3*x)/6 + x*sin(2*x)/2