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Chap7_Sec1

# 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.
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
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.
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.
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
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
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 + =
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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