f(t) = 3 + sin (\frac{2πt}{5})

sin (\frac{2π}{P} t) = sin(ωt)

\begin{aligned}A_{0} & = 3 \\ B_{1} & = 1 \end{aligned}

A_{0} = \frac{1}{P} ∫_{0}^{P} f(t)dt = \frac{1}{5} ∫_{0}^{5} [3 + sin (\frac{2πt}{5})] dt

\frac{1}{5} ∫_{0}^{5} sin (\frac{2πt}{5}) dt = \frac{1}{5} (\frac{5}{2π} ) [− cos (\frac{2πt}{5})]_{0}^{5}

\frac{1}{2π} [− cos(2π) − (− cos(0))] = \frac{1}{2π} [−1 + 1] = 0

A_{0} = 3 + 0 = 3

A_{1} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2πt}{P}) dt =  \frac{2}{5} ∫_{0}^{P} [3 + sin (\frac{2πt}{5})] cos (\frac{2πt}{5}) dt
B_{1} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2πt}{P}) dt =  \frac{2}{5} ∫_{0}^{P} [3 + sin (\frac{2πt}{5})] sin (\frac{2πt}{5}) dt

A_{1} = \frac{2}{5} ∫_{0}^{P} 3 cos (\frac{2πt}{5}) dt + \frac{2}{5} ∫_{0}^{P} sin (\frac{2πt}{5}) cos (\frac{2πt}{5}) dt

A_{1} =  \frac{6}{5} (\frac{5}{2π} ) sin [\frac{2πt}{5}]_{0}^{5} =  \frac{6}{2π} [sin(2π) − sin(0)] = 0

A_{1} = 0

B_{1} = \frac{2}{5} ∫_{0}^{5} 3 sin (\frac{2πt}{5}) dt + \frac{2}{5} ∫_{0}^{5} sin (\frac{2πt}{5}) sin (\frac{2πt}{5}) dt

B_{1} = \frac{6}{5} (−\frac{5}{2π} ) cos [\frac{2πt}{5}]_{0}^{5} + \frac{2}{5} ∫_{0}^{5} \frac{1}{2} [1 − cos (\frac{4πt}{5})] dt

B_{1} = \frac{6}{5} [cos(2π) − cos(0)] + \frac{1}{5} ∫_{0}^{5} dt − \frac{1}{5} ∫_{0}^{5} cos (\frac{4πt}{5}) dt

B_{1} =  0 +  1 − \frac{1}{5} (\frac{5}{4π} ) sin [\frac{4πt}{5}]_{0}^{5}

B_{1} =  0 +  1 − \frac{1}{5} (\frac{5}{4π} ) [sin(4π) − sin(0)]

B_{1} =  0 +  1 − 0 = 1

f(t) = \{{1  0 ≤ t ≤ 1 \atop 0  1 ≤ t ≤ 2}

A_{0} = \frac{1}{P} ∫_{0}^{P} f(t)dt = \frac{1}{2} ∫_{0}^{1}(1)dt + \frac{1}{2} ∫_{1}^{2}(0)dt

A_{1} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2π}{P} t) dt = ∫_{0}^{1} 1 cos(πt)dt + ∫_{1}^{2} 0 cos(πt)dt

B_{1} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2π}{P} t) dt = ∫_{0}^{1} 1 sin(πt)dt + ∫_{1}^{2} 0 sin(πt)dt

A_{1} = \frac{1}{π} sin[πt]_{0}^{1} = \frac{1}{π} (0 − 0) = 0

B_{1} = −\frac{1}{π} cos[πt]_{0}^{1} = −\frac{1}{π} (−1 − (1)) = −\frac{1}{π} (−2) = \frac{2}{π}

A_{2} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2π · 2}{P}t) dt = ∫_{0}^{1} 1 cos(2πt)dt + ∫_{1}^{2} 0 cos(2πt)dt

B_{2} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2π · 2}{P}t) dt = ∫_{0}^{1} 1 sin(2πt)dt + ∫_{1}^{2} 0 sin(2πt)dt

A_{3} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2π · 3}{P}t) dt = ∫_{0}^{1} 1 cos(3πt)dt + ∫_{1}^{2} 0 cos(3πt)dt

B_{3} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2π · 3}{P}t) dt = ∫_{0}^{1} 1 sin(3πt)dt + ∫_{1}^{2} 0 sin(3πt)dt

A_{2} = \frac{1}{2π} sin[2πt]_{0}^{1} = \frac{1}{2π} (0 − 0) = 0

B_{2} = −\frac{1}{2π} cos[2πt]_{0}^{1} = −\frac{1}{2π} (1 − 1) = 0

A_{3} = \frac{1}{3π} sin[3πt]_{0}^{1} = \frac{1}{3π} (0 − 0) = 0

B_{3} = −\frac{1}{3π} cos[3πt]_{0}^{1} = −\frac{1}{3π} (−1 − (1)) = \frac{2}{3π}

f(t) ∼ \frac{1}{2} + \frac{2}{π} sin(πt) + \frac{2}{3π} sin(3πt)

A_{4} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2π · 4}{P}t) dt = ∫_{0}^{1} 1 cos(4πt)dt + ∫_{1}^{2} 0 cos(4πt)dt

B_{4} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2π · 4}{P}t) dt = ∫_{0}^{1} 1 sin(4πt)dt + ∫_{1}^{2} 0 sin(4πt)dt

A_{5} = \frac{2}{P} ∫_{0}^{P} f(t) cos (\frac{2π · 5}{P}t) dt = ∫_{0}^{1} 1 cos(5πt)dt + ∫_{1}^{2} 0 cos(5πt)dt

B_{5} = \frac{2}{P} ∫_{0}^{P} f(t) sin (\frac{2π · 5}{P}t) dt = ∫_{0}^{1} 1 sin(5πt)dt + ∫_{1}^{2} 0 sin(5πt)dt

A_{4} = \frac{1}{4π} sin[4πt]_{0}^{1} = \frac{1}{4π} (0 − 0) = 0

B_{4} = −\frac{1}{4π} cos[4πt]_{0}^{1} = −\frac{1}{4π} (1 − 1) = 0

A_{5} = \frac{1}{5π} sin[5πt]_{0}^{1} = \frac{1}{5π} (0 − 0) = 0

B_{5} = −\frac{1}{5π} cos[5πt]_{0}^{1} = −\frac{1}{5π} (−1 − (1)) = \frac{2}{5π}

f(t) ∼ \frac{1}{2} + \frac{2}{π} sin(πt) + \frac{2}{3π} sin(3πt) + \frac{2}{5π} sin(5πt)