Compute area of region bounded by curves y=tanx and y=tan^2 x (-π/2<x<π/2)

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Solution

tan x = tan^2 x
tan x( 1 - tan x) = 0
tan x = 0
tan x = 1
x=0; x=pi/4

Area =
Area = ∫ (tan x - tan^2 x) dx

= ∫tan x dx - ∫tan^2 x dx
= -ln (cos x) - ∫(sec^2 x -1) dx
= -ln (cos x) - ∫ sec^2 x dx +∫ dx
= -ln (cos x) - tan x + x

Integrate from 0 to pi/4
F(x) = -ln (cos x) - tan x + x
F(pi/4) = -ln (cos pi/4) - tan pi/4 + pi/4
F(0) = 0
F(pi/4)-F(0) = -ln (cos pi/4) - tan pi/4 + pi/4
F(pi/4)-F(0) = -ln (1/sqrt(2)) - 1 + pi/4
= -ln (2^(-1/2)) - 1 + pi/4
= (1/2) ln(2) - 1 + pi/4

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