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 Proofs: Integral sin, cos, sec2, csc cot, sec tan, csc2 (Math | Calculus | Integrals | Table Of | ResultTrig)

 Discussion of cos x dx = sin x + C  sin x dx = -cos x + C  sec2 x dx = tan x + C  csc x cot x dx = -csc x + C  sec x tan x dx = sec x + C  csc2 x dx = -cot x + C
1. Proofs

For each of these, we simply use the Fundamental of Calculus, because we know their corresponding derivatives.
cos(x) =  sin(x), cos(x) dx = sin(x) + c
-sin(x) =  cos(x), sin(x) dx = -cos(x) + c
sec^2(x) =  tan(x), sec^2(x) dx = tan(x) + c
-csc(x)cot(x) =  csc(x), csc(x)cot(x) dx = -csc(x) + c
sec(x)tan(x) =  sec(x), sec(x)tan(x) dx = sec(x) + c
-csc^2(x) =  cot(x), csc^2(x) dx = -cot(x) + c

sin(x) = cos(x) cos(x) = -sin(x) tan(x) = sec2(x),
csc(x) = -csc(x)cot(x) sec(x) = sec(x)tan(x) cot(x) = -csc2(x).