Jika f(x) = 1+sin x + sin^2 + sin^3 + ...., 0° <= x < 45°, maka integral 0 sampai 45° f(x) dx = .... A. -√2 B. -1 C. 0 D. 1 E. √2

f(x) = 1 + sin x + sin^2 x + sin^3 x + .... = S~ = a/(1 - r)
a = 1, r = sin x
f(x) = a/(1 - r)
= 1/(1 - sin x)
= 1/(1 - sin x) . (1 + sin x)/(1 + sin x)
= (1 + sin x)/(1 - sin^2 x)
= (1 + sin x)/(cos^2 x)
= 1/(cos^2 x) + (sin x)/(cos^2 x)
= sec^2 x + (sin x)/(cos x) . 1/(cos x)
= sec^2 x + Tan x sec x
Int f(x) dx
= Int (sec^2 x + Tan x sec x) dx
= tan x + sec x | batas x = 0° sampai x = 45°
= (Tan 45° + sec 45°) - (Tan 0° + sec 0°)
= (1 + √2) - (0 + 1)
= 1 + √2 - 1
= √2