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 termseverything 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 d =- ln | cos | + C (Dont forget the minus), R sec d = ln | sec + tan | + 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, its 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 whats left (the dv ) is not too bad to integrate....
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