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(a)Prove  that2a0f(x)dx=2a0f(X)dx,if f(2ax)=f(x) and evaluate2π0cos5xdx
 if f(2ax)=f(x)

(b) Find the values of a and b such that the function defined by

f(x)=5,if x2ax+bif 2<x<10 is a continous function21,if x10


Solution

(a)2a0f(x)dx=a0f(x)dx+2aaf(x)dx         For derivationsLetI1=2a0f(x) dx     For ProblemPut 2ax=tx=2at dx=dtx=at=a,x=2at=0I=0af(2at)(dt)=0af(2ax)dx(b)f(x)is continuousltx2f(x)=ltx2+andltx10f(x)ltx10+f(x)5=2a+b and 10a+b=21a=2,b=1

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