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Question

The area of the region bounded by the curve y=tanx, the tangent to the curve at x=π4 and the x-axis is

A
14(loge41)
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B
14(loge21)
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C
12(loge41)
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D
12(loge21)
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Solution

The correct option is A 14(loge41)

Slope of the tangent at x=π4 is sec2π4=2
Equation of the tangent at A is,
y1=2(xπ4)
Coordinates of B are (π412,0)
BC=12

Required area
= Area of OACO Area of ΔABC
=π/40tanx dx12×AC×BC
=[logesecx]π/4012×1×12=(loge20)14=14(loge41)

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