Integral sin, cos, sec^2, -csc cot, sec tan, -csc^2

Dave's Math Tables: 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

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).