Antiderivatives are just the reverse of derivatives or differentials. Typically, since d(x^n)=nx^(n-1). The antiderivative of nx^(n-1) is therefore x^n. Or, to put it another way, the antiderivative of x^n is (x^(n+1))/(n+1).
The derivative of sin x is cos x, and the derivative of cos x is -sin x; so the antiderivative of cos x is sin x and of sin x is -cos x. There are many more examples in trigonometry. Methods of finding the antiderivates of trig functions also tend to utilise trig identities.
Antiderivatives are part of the integration process.