AP CALCULUS NOTES SECTION 8.2 INTEGRATION BY PARTS The u-substitution method of integration that we previously examined is based on the chain rule for differentiation. Integration by parts, which we examine in this section, is a method that is based on the product rule for differentiation. A.) Integration by Parts: Let u and v represent differentiable functions of x. d uv dx u dv uv v du Ex.1.) Find the x sin x dx . Note: To use integration by parts successfully, the choice of u and v must be made so that the new integral is easier than the original. When the integrand consists of a product of two different types of functions, let u designate the function that appears first in the acronym LIATE and let dv denote the rest of the integrand. Ex.2.) x ln x dx Repeated Integration by Parts: Ex.3.) x 2 e x dx Ex.4.) e x sin x dx Performing Repeated Integration by Parts using Tabular Integration – one of the functions in the integrand is a polynomial. Steps for Tabular Integration: 1.) Repeatedly differentiate the polynomial (left column) until you get a 0 entry. 2.) Repeatedly antidifferentiate the other function (right column). 3.) Connect the columns with diagonal segments that alternate signs from + to – . Ex.5.) x3 sin x dx C.) Integration by Parts for Definite Integrals: Ex.6.) 1 0 b a u dv uv a v du b b a tan 1 x dx Ex.7.) Find the area of the region bounded by the curve y xe x and the x-axis on 0, 4 .