Antiderivatives and indefinite integrals

Learning objectives

Define an antiderivative, explain why the indefinite integral includes an arbitrary constant + C, and verify an antiderivative by differentiating it

Apply the power rule for antiderivatives, ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1, together with the constant multiple, sum and difference rules, after rewriting radicals, reciprocals and products in power form

Recall the antiderivatives that reverse the standard derivatives: 1/x, eˣ, sin x, cos x, sec²x, sec x tan x, csc²x, csc x cot x, and the forms that give arcsin x and arctan x

Find a particular antiderivative from an initial condition, for example a position function from a velocity function and an initial position

Simplify an integrand algebraically before integrating, for example by expanding a product or dividing each term of a quotient by a monomial