Let f(x)=x∫0et(lnsect−sec2t)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 Ceπ/3ln2 f(x)=x∫0et(lnsect−sec2t)dt =x∫0et(lnsect−sec2t+tant−tant)dt [ddt(lnsect)=tant] =x∫0et(lnsect+tant)dt−x∫0et(tant+sec2t)dt =[et(lnsect)]x0−[ettant]x0
∴f(x)=ex(lnsecx−tanx)
Required area =π/3∫0(f(x)−g(x))dx =π/3∫0(ex(lnsecx−tanx)+2extanx)dx =π/3∫0ex(lnsecx+tanx)dx =[exlnsecx]π/30=eπ/3ln2