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Question

Let gi:[π8,3π8]R,i=1,2 and f:[π8,3π8]R be functions such that g1(x)=1,g2(x)=|4xπ| and f(x)=sin2x, for all x[π8,3π8].
Define Si=3π8π8f(x)gi(x)dx, i=1,2

The value of 48S2π2 is

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Solution

S2=3π8π8sin2x|4xπ|dx
=3π8π84sin2x.xπ4dx
Let xπ4=tdx=dt
S2=π8π84sin2(π4+t)|t|dt=π8π82[1cos(2(π4+t))]|t| dt=π8π8(2+2sin2t)|t|dt=π8π82|t|dt+2π8π8|t|sin(2t)dt
=4π80tdt+0
S2=2t2|π/80=π232
48S2π2=32


Alternate Method
S2=3π8π8sin2x|4xπ|dx (1)

S2=3π8π8sin2(π8+3π8x)4(π8+3π8x)πdx

S2=3π8π8cos2x|4xπ|dx (2)

From (1)+(2)
2S2=3π8π8|4xπ|dx

2S2=π4π8(π4x)dx+3π8π4(4xπ)dx

2S2=[πx2x2]π/4π/8+[2x2πx]3π/8π/4

2S2=π216

Now
48S2π2=32

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