Question

The area of the region above the x-axis bounded by the curve y =  tanx, 0 ≤ x ≤ π/2  and the tangent to the curve at x = π/4 is

(A) 1/2(log2 - 1/2)

(B) 1/2(log2 + 1/2)

(C) 1/2(1 - log2)

(D) 1/2(1 + log2)

Answer 1

Answer is (A) 1/2(log2 - 1/2)

See Fig.

Required area = ∫tanx - area under tangentat (π/4, 1)   (1)

Now slope of tangent is d/dxtanx at x = π/4 = sec²x|_{at x=π/4} = 2

Therefore, equation of tangent is y - 1 = 2(x - π/4) or y = 2x + (1 - π/2)

This tangent cuts x-axis when y = 0 

Therefore,