Question

cos theta (integration)​

Answer 1

Step-by-step explanation:

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)