Math2.org Math Tables: Integral sin, cos, sec^2, csc cot, sec tan, csc^2

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
 

See also:
  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).