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Question

If g(x)=x0(|sint|+|cost|) dt, then the value of g(x+nπ2),( where nN) is

A
g(x)+3n
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B
g(x)+n
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C
g(x)+2n
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D
g(x)+n2
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Solution

The correct option is C g(x)+2n
We have,
g(x)=x0(|sint|+|cost|) dt
Now,
g(x+nπ2)=x+nπ20(|sint|+|cost|)dt=x0(|sint|+|cost|)dt+x+nπ2x(|sint|+|cost|)dt
Using the property:
a+nTaf(x) dx=nT0f(x) dx
g(x+nπ2)=g(x)+nπ20(|sint|+|cost|)dt
As (|sint|+|cost|) has a period π2
g(x+nπ2)=g(x)+g(nπ2)g(x+nπ2)=g(x)+nπ20(|sinx|+|cosx|)dxg(x+nπ2)=g(x)+nπ20(sinx+cosx)dxg(x+nπ2)=g(x)+n[sinxcosx]π20g(x+nπ2)=g(x)+2n

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