∫ -cos(x) dx = -sin(x) + C
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-(1/2)X^2 [negative half X squared]
Cosine to the negative first power and cosine cancel each other out because cosine to the negative first power is one over cosine, and one over anything times anything is just one.
Because the cosine of some angles is positive and the cosine of some other angles is negative.
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
An integral and an anti-derivative are the same thing. Integration means the process of finding the integral, just as anti-differentiation means the process of finding the anti-derivative.