notes Overview of Integration Techniques

Thomas' Calculus: Early Transcendentals

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Unformatted text preview: Overview of Integration Techniques MAT 104 – Frank Swenton, Summer 2000 Fundamental integrands (see table, page 400 of the text) • Know well the antiderivatives of basic terms–everything reduces to them in the end. • Remember to think of 1 x a as x- a when antidifferentiating with the power rule. • A few other useful integrals to know: R tan θ dθ =- ln | cos θ | + C (Don’t forget the minus), R sec θ dθ = ln | sec θ + tan θ | + C , and Z dw a 2 + w 2 = 1 a tan- 1 w a + C (Note the a ’s vs. a 2 ’s, and that the w 2 has coefficient 1) Substitution (make sure you substitute for all components, including the dx ) • When using substitution for definite integrals, be very careful with the limits of integration ! • Be sure to account for each term in the integral when substituting, especially the “ dx ”. • For indefinite integrals, be sure that your final answer is in terms of the variable that was originally given. • When making substitutions involving fractional powers, it’s often easier to reverse the substitution (e.g., instead of w = √ x , dw = 1 2 √ x dx ; use x = w 2 , dx = 2 w dw ). Integration by parts : R udv = uv- R v du • Factor the integrand so that one factor (the u ) becomes simpler when differentiated and what’s left (the dv ) is not too bad to integrate....
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