Antiderivatives of Trigonometric and Exponential Functions

Every antiderivative rule here is just a derivative rule read in reverse. You already know d/dx[sin(x)] = cos(x), so the antiderivative of cos(x) is sin(x). You know d/dx[e^x] = e^x, so e^x antidifferentiates to itself. The one that trips people up is sine: because d/dx[cos(x)] = -sin(x), the derivative of cosine already carries a negative sign. To undo that sign and land back on plain sin(x), the antiderivative of sin(x) has to be -cos(x). Check it: d/dx[-cos(x)] = -(-sin(x)) = sin(x). The sign flip isn't a rule to memorize blindly, it's forced by what makes the derivative come out right.

The same logic covers the other trig derivatives you learned earlier in the course. Since d/dx[tan(x)] = sec²(x), the antiderivative of sec²(x) is tan(x). Since d/dx[csc(x)] = -csc(x)cot(x), the antiderivative of csc(x)cot(x) is -csc(x). Every entry on this list is a flipped version of a derivative you already trust, so when in doubt, differentiate your answer and see if you get back the original function.

This topic plugs directly into evaluating definite integrals with the Fundamental Theorem of Calculus: once you have an antiderivative, you evaluate it at the bounds and subtract. It also sets up initial value problems, where a condition like F(0) = 0 pins down the constant C that's otherwise free. Trig and exponential antiderivatives keep reappearing once substitution and more complex integrands show up later in the unit, so having these reversed rules automatic saves real time.