Indefinite Integrals and the Net Change Theorem Recall that the indefinite integral is the set of all possible antiderivatives of a given function, denoted by ∫+= CxFdxxf )()( where F is any antiderivative of f , i.e., F ′(x) = f (x), and C is an arbitrary number. Definite vs. Indefinite Integrals Despite the similar names and notations, and their close relation (via the Fundamental Theorem of Calculus), definite and indefinite integrals are objects of quite different nature. A definite integral represents a number, while an indefinite is a function (or, rather, the general form of a family of functions). This is an important fact to keep in mind. Ex. Cxxxdxxx ++−=+−∫32)345(254 , but 30030)000()6832(32)345(2025204=−=+−−+−=+−=+−∫xxxdxxx A little bit of the Chain Rule…. Ex. Because )cos()sin(1kxkxkdxd= and )sin()cos(1kxkxkdxd−=. Therefore, if k ≠ 0, then ∫+= Ckxkdxkx )sin(1)cos(, and ∫+−= Ckxkdxkx )cos(1)sin(Table of some common indefinite integrals: []∫∫∫±=± dxxgdxxfdxxgxf )()()()( ∫∫= dxxfcdxxfc )()( ∫+= Ckxdxk , for all numbers k. A special case: when k = 0, CCxdx =+=∫00 ∫++=+Cxndxxnn 111, for all numbers n, n ≠ −1. For any number k ≠ 0, ∫+= Ckxkdxkx )sin(1)cos( ∫+−= Ckxkdxkx )cos(1)sin( ∫+= Ckxkdxkx )tan(1)(sec2 ∫+= Ckxkdxkxkx )sec(1)tan()sec( ∫+−= Ckxkdxkx )cot(1)(csc2 ∫+−= Ckxkdxkxkx )csc(1)cot()csc(The Net Change Theorem Theorem: The integral of a rate of change is the net change: ∫−=′baaFbFdxxF )()()(. Therefore, for example, if v(t) is the velocity and s(t) is the position of an object, then the definite integral ∫−=baasbsdttv )()()( represents the net distance traveled between t = a and t = b. Ex. The velocity of an oscillating mass-spring system is described by v(t) = cos t. Find the net distance traveled between (i.) t = 0 and t = π/2, and (ii.) t = 2π and t = 6π. (i.) 101)0sin()2/sin(sincos2/02/0=−=−==∫πππtdtt (ii.)