Lesson 2- Generalized Power Formulas

# Lesson 2- Generalized Power Formulas -...

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ANTIDERIVATIVES (INTEGRAL)

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GENERALIZED POWER FORMULA (Integration by Simple Substitution) identify an integrand that can be integrated using simple substitution; perform integration using the generalized power formula; relate integration by power formula to the generalized integration formula; and consider and use the “introduction of neutralizing/correction factor” as an alternative technique of integration. OBJECTIVES:
A technique called substitution , that can often be used to transform complicated integration problems into simpler ones. NTEGRATION BY SUBSTITUTION The method of substitution can be motivated by examining the chain rule from the viewpoint of antidifferentiation. For this purpose, suppose that F is an antiderivative of f and that g is a differentiable function. The chain rule implies that the derivative of F(g(x)) can be expressed as which we can write in integral form as [ ] ) ( ' )) ( ( ' )) ( ( x g x g F x g F dx d = + = C x g F dx x g x g F )) ( ( ) ( ' )) ( ( '

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Or since F is an antiderivative of f, For our purposes it will be useful to let u=g(x) and to write the differential form . Thus, + = C x g F dx x g x g f )) ( ( ) ( ' )) ( ( ) ( ' x g dx du = dx x g du ) ( ' =

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