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lecture30hout - MATH 006 Calculus and Linear...

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MATH 006 Calculus and Linear Algebra (Lecture 30) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 11
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Outline 1 Antiderivatives 2 Indefinite Integrals 3 Indefinite Integrals of Basic Functions Albert Ku (HKUST) MATH 006 2 / 11
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Antiderivatives Antiderivatives Definition A function F is an antiderivative of a function f if F ( x ) = f ( x ) . Example Find an antiderivative of f ( x ) = 2 x . Solution Since ( x 2 ) = 2 x , x 2 is an antiderivative. In fact, there are many antiderivatives of 2 x , such as x 2 + 1 and x 2 + 2. Albert Ku (HKUST) MATH 006 3 / 11
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Antiderivatives We have the following useful theorem about antiderivatives: Theorem If the derivatives of two functions are equal, that is F ( x ) = G ( x ) , then the functions differ by a constant, that is F ( x ) = G ( x ) + C for some constant C. Proof. Since F ( x ) = G ( x ), ( F ( x ) - G ( x )) = 0. It is obvious that a function must be a constant if its derivative is zero. Therefore, there exists a constant k such that F ( x ) - G ( x ) = C . Hence F ( x ) = G ( x ) + C .
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