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Math 106: Review for Exam I INTEGRATION TIPS * Substitution: usually let w = an inside function, especially if w’ is also present in the integrand © Parts: fra uv — [va or [wi ae = we ~ f vods How to choose which part is u? Let u be the part that is higher up in the LLATE mnemonic below. (The mnemonics ILATE and LIPET will work equally well if you have learned one of those instead; in the latter A is replaced by P, which stands for “polynomial”.) Logarithms (such as In) Inverse trig (such as arctan, arcsin x.) Algebraic (such as a, x?, x? + 4) Trig (such as sina, cos 227) Exponentials (such as ¢”, e**) « Rational Functions (one polynomial divided by another): if the degree of the numerator is greater than or equal to the degree of the numerator, do long division then integrate the result. Partial Fractions: here’s an illus ‘ative example of the setup. 3a? +411 page B a Cc , Dare (g+1)(@—3)(@? +5) 2@+1"a—-3 (@—3)? a?+5 Each linear term in the denominator on the left gets a constant above it on the right; the squared linear factor (2 — 3) on the left appears twice on the right, once to the second power. Each irreducible quadratic term on the left gets a linear term (Da + E here) above it on the right. 1. Find the following. yk xed ee -[e Sta = gel oe oe ee 1 Subiw= dk => dur ik ‘dy _ ak = 2dwe Te S & Si el wy fo d : Bl] Et Parés: Usinx dvz* ay 4 =a due aoe Ne z Partial Pavbial Fractrng 5 af ieee = es Feta = =o eI de ele x exec 5 Cele xs K+] St K-1b = ALC sao 3) Lek x23 Wx Alo €(GKEC)-O FAHD KFO3-O= 2-1 +@otcl-3) 3 a ee ee = 4 tn(+( = $ In(x*(] Kel DP ~G=2-2 +Bl4s)C > -2 Eee fo ase HS table | e/ , lL rr Fla ts ae: = 
