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Proofs: Integral sin, cos, sec2, csc cot, sec tan, csc2
(Math | Calculus | Integrals | Table Of | ResultTrig)
 
Discussion of
(integral)cos x dx = sin x + C 
(integral)sin x dx = -cos x + C 
(integral)sec2 x dx = tan x + C 
(integral)csc x cot x dx = -csc x + C 
(integral)sec x tan x dx = sec x + C 
(integral)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) = (d/dx) sin(x), (integral)cos(x) dx = sin(x) + c
-sin(x) = (d/dx) cos(x), (integral)sin(x) dx = -cos(x) + c
sec^2(x) = (d/dx) tan(x), (integral)sec^2(x) dx = tan(x) + c
-csc(x)cot(x) = (d/dx) csc(x), (integral)csc(x)cot(x) dx = -csc(x) + c
sec(x)tan(x) = (d/dx) sec(x), (integral)sec(x)tan(x) dx = sec(x) + c
-csc^2(x) = (d/dx) cot(x), (integral)csc^2(x) dx = -cot(x) + c
 

See also:
 (d/dx) sin(x) = cos(x)(d/dx) cos(x) = -sin(x)(d/dx) tan(x) = sec2(x),
(d/dx) csc(x) = -csc(x)cot(x)(d/dx) sec(x) = sec(x)tan(x)(d/dx) cot(x) = -csc2(x).

  
 
  

 
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