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lecture30

# lecture30 - Find(a Z x 2 dx(b Z t 2 dt(c Z y 2 dy Notice...

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Math 006 (Lecture 30) Antiderivatives A Function F is an antiderivative of a function f if F ( x ) = f ( x ) . Example 1. Find an antiderivative of f ( x ) = 2 x . Theorem 1. If the derivatives of two functions are equal, that is F ( x ) = G ( x ) , then the functions differ by at most a constant, that is F ( x ) = G ( x ) + k for some constant k . Example 2. Let f ( x ) = 3 x 2 . (a) Find all antiderivatives of f ( x ). (b) Find the antiderivative of f ( x ) that passes through the point (0 , 0); through (0 , 1); through the point (0 , 2). 1

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Indefinite Integral We use the symbol f ( x ) dx, called the indefinite integral , to represent the family of all antiderivatives of f ( x ) and write f ( x ) dx = F ( x ) + C if F ( x ) = f ( x ) . The function f ( x ) is called the integrand . The symbol dx indicate that the antiderivative is performed with respect to the variable x . The arbitrary constant C is called the
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Unformatted text preview: Find (a) Z x 2 dx (b) Z t 2 dt (c) Z y 2 dy Notice that indeﬁnite integration and diﬀerentiation are reserve operations , except for the addition of the constant of integration. The action can be expressed as d dx ±Z f ( x ) dx ² = f ( x ) Z F ( x ) dx = F ( x ) + C. Indeﬁnite Integrals of Basic Functions (a) Z x n dx = x n +1 n + 1 + C (b) Z e x dx = e x + C (c) Z 1 x dx = ln | x | + C (d) Z kf ( x ) dx = k Z f ( x ) dx for any constant k , (e) Z [ f ( x ) ± g ( x )] dx = Z f ( x ) dx ± Z g ( x ) dx 2 Example 4. Find each indeﬁnite integral: (a) Z 2 dx (b) Z 16 e t dt (c) Z 3 x 4 dx (d) Z (2 x 5-3 x 2 + 1) dx (e) Z ± 5 x-4 e x ² dx (f) Z ± 2 x 2 / 3-3 x 4 ² dx 3...
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lecture30 - Find(a Z x 2 dx(b Z t 2 dt(c Z y 2 dy Notice...

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