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Question

Let f(x)=x0et(lnsectsec2t)dt and g(x)=2extanx. Then, the area bounded by the curves y=f(x) and y=g(x) between the ordinates x=0 and x=π3 is

A
12eπ/3ln2
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B
12eπ/3ln4
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C
eπ/3ln2
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D
None of these
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Solution

The correct option is C eπ/3ln2
f(x)=x0et(lnsectsec2t)dt
=x0et(lnsectsec2t+tanttant)dt
[ddt(lnsect)=tant]
=x0et(lnsect+tant)dtx0et(tant+sec2t)dt
=[et(lnsect)]x0[ettant]x0

f(x)=ex(lnsecxtanx)


Required area =π/30(f(x)g(x))dx
=π/30(ex(lnsecxtanx)+2extanx)dx
=π/30ex(lnsecx+tanx)dx
=[exlnsecx]π/30=eπ/3ln2

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