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Chap7_Sec1 - 7 TECHNIQUES OF INTEGRATION TECHNIQUES OF...

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7 TECHNIQUES OF INTEGRATION TECHNIQUES OF INTEGRATION
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TECHNIQUES OF INTEGRATION Due to the Fundamental Theorem of Calculus (FTC), we can integrate a function if we know an antiderivative, that is, an indefinite integral. We summarize the most important integrals we have learned so far, as follows.
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FORMULAS OF INTEGRALS 1 1 ( 1) ln | | 1 ln n n x x x x x x dx C n dx x C n x a e dx e C a dx C a + = + ≠ - = + + = + = +
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2 2 sin cos cos sin sec tan csc cot sec tan sec csc cot csc x dx x C x dx x C dx x C dx x C x xdx x C x x dx x C = - + = + = + = - + = + = - + FORMULAS OF INTEGRALS
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1 1 2 2 2 2 sinh cosh cosh sinh tan ln | sec | cot ln | sin | 1 1 1 tan sin xdx x C xdx x C x dx x C xdx x C x x dx C dx C x a a a a a x - - = + = + = + = + = + = + ÷ ÷ + - FORMULAS OF INTEGRALS
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TECHNIQUES OF INTEGRATION In this chapter, we develop techniques for using the basic integration formulas. This helps obtain indefinite integrals of more complicated functions.
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TECHNIQUES OF INTEGRATION We learned the most important method of integration, the Substitution Rule, in Section 5.5 The other general technique, integration by parts, is presented in Section 7.1
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TECHNIQUES OF INTEGRATION Then, we learn methods that are special to particular classes of functions—such as trigonometric functions and rational functions.
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TECHNIQUES OF INTEGRATION Integration is not as straightforward as differentiation. There are no rules that absolutely guarantee obtaining an indefinite integral of a function. Therefore, we discuss a strategy for integration in Section 7.5
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7.1 Integration by Parts In this section, we will learn: How to integrate complex functions by parts. TECHNIQUES OF INTEGRATION
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Every differentiation rule has a corresponding integration rule. For instance, the Substitution Rule for integration corresponds to the Chain Rule for differentiation. INTEGRATION BY PARTS
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The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts . INTEGRATION BY PARTS
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The Product Rule states that, if f and g are differentiable functions, then INTEGRATION BY PARTS [ ] ( ) ( ) ( ) '( ) ( ) '( ) d f x g x f x g x g x f x dx = +
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In the notation for indefinite integrals, this equation becomes or INTEGRATION BY PARTS [ ] ( ) '( ) ( ) '( ) ( ) ( ) f x g x g x f x dx f x g x + = ( ) '( ) ( ) '( ) ( ) ( ) f x g x dx g x f x dx f x g x + =
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We can rearrange this equation as: INTEGRATION BY PARTS ( ) '( ) ( ) ( ) ( ) '( ) f x g x dx f x g x g x f x dx = - Formula 1
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