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MathCalculus

Find the Most General Antiderivative or Indefinite Integral of 8 sec x (sec x − 4 tan x)

Expanding the product gives 8 sec²x − 32 sec x tan x, and both terms are standard antiderivative table entries. The most general antiderivative is F(x) = 8 tan x − 32 sec x + C, which returns the original integrand when differentiated.

Question

For the following exercises, find the general form for the antiderivative F(x)F(x) of each function f(x)f(x).

  1. f(x)=8sec⁡x (sec⁡x−4tan⁡x)f(x)=8\sec x\,(\sec x-4\tan x)

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Evidence boundary

This page works the function published as exercise 488 in the OpenStax Calculus Volume 1 §4.10 exercise set, whose instruction is to find the general form of the antiderivative F(x) of each function f(x). The expansion, the term-by-term integration and the differentiation check are carried out on that same function; the page does not claim the problem comes from any other worksheet, textbook or answer key. Because the request is for a general form rather than a single function, the constant C is left undetermined and the formula is stated as valid on intervals that avoid the points where cos x = 0; the page does not claim a particular value of C.