notes Overview of Integration Techniques - Overview of Integration Techniques MAT 104 Frank Swenton Summer 2000 Fundamental integrands(see table page

# Thomas' Calculus: Early Transcendentals

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Overview of Integration TechniquesMAT 104 – Frank Swenton, Summer 2000Fundamental integrands(see table, page 400 of the text)Know wellthe antiderivatives of basic terms–everything reduces to them in the end.Remember to think of1xaasx-awhen antidifferentiating with the power rule.A few other useful integrals to know:Rtanθ dθ=-ln|cosθ|+C(Don’t forget the minus),Rsecθ dθ= ln|secθ+ tanθ|+C, andZdwa2+w2=1atan-1wa+C(Note thea’s vs.a2’s, and that thew2has coefficient 1)Substitution(make sure you substitute for all components, including thedx)When using substitution for definite integrals, be very careful with thelimits of integration!Be sure to account foreach termin the integral when substituting, especially the “dx”.For indefinite integrals, be sure that your final answer is in terms of the variable that was originallygiven.When making substitutions involving fractional powers, it’s often easier to reverse the substitution(e.g., instead ofw=x,dw=12xdx; usex=w2,dx= 2w dw).Integration by parts:Ru dv=uv-Rv duFactor the integrand so that one factor (theu) becomes simpler when differentiated and what’s left(thedv) is not too bad to integrate.