Find the area between the curve of f(x) = (tan x)/(sec x + tan x) and the x-axis from x=0 to x=π.

Question

Find the area between the curve of f(x) = (tan x)/(sec x + tan x) and the x-axis from x=0 to x=π.

Answer 1

f(x) = tan(x)/(sec(x) + tan(x))

Before integrating, simplify f(x)

tan(x)(sec(x)-tan(x))/(sec²(x) - tan²(x)) = (tan(x)sec(x) - tan²(x))/1 = tan(x)sec(x) - (sec²(x) - 1)

A = ∫₀^π tan(x)sec(x) - sec²(x) + 1 dx

= sec(x) -tan(x) + x |₀^π = -1 - 0 + π -1 + 0 - 0

A = π-2

Answer 2

Area A = ∫₀^πtanx/(tanx + secx)dx = ∫₀^πsinx/(sinx + 1)dx = ∫₀^π(1 - 1/(sinx + 1)dx = ∫₀^π(1 - (1 - sinx)/cos²x)dx =

=∫₀^π(1 - sec²x + sinx/cos²x)dx = (x - tanx + secx)₀^π = (π - tanπ + secπ) - (0 - tan0 + sec0) = π - 1 - 1 = π - 2