Antiderivatives, indefinite integrals and substitution

Learning objectives

Explain why any two antiderivatives of a function differ only by a constant, verify a proposed antiderivative by differentiating it, and apply the power rule together with the constant multiple and sum rules after rewriting radicals, reciprocals and expanded products in power form

Recall the antiderivatives that reverse the standard derivatives, including 1/x, eˣ, sin x, cos x, sec²x, sec x tan x, and the forms giving arcsin and arctan

Determine a particular antiderivative from an initial condition, for example recovering a position function from a velocity function and a known starting position

Recognize an integrand of the form f(g(x))g′(x) and reverse the chain rule with the substitution u = g(x), adjusting for a missing constant factor

Evaluate a definite integral by substitution, either by converting the limits to values of u or by returning to the original variable before substituting the original limits

Prepare an integrand for integration by long division when the numerator has degree at least that of the denominator, or by completing the square to reach an arctan or arcsin form