Appears Twice - Calculus - Solved Exam, Exams of Calculus

Download Appears Twice - Calculus - Solved Exam and more Exams Calculus in PDF only on Docsity!

Math 106: Review for Exam II INTEGRATION TIPS ° Substitution: usually let w = an inside function, especially if w’ is also present in the integrand © Parts: [vao=wo— fvaw or fwide=w— fwode How to choose which part is u? Let v be the part that is higher up in the LIATE 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 x, arcsin x) Algebraic (such as a, #”,#° + 4) Trig (such as sin x, cos 2) Exponentials (such as e*, 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 illustrative example of the setup. 30? +11 peat eee fC Dae (p+ 1)(@—38)(2274+5) +1 2-3 (@-3) ° 2245 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. ‘Trigonometric Antiderivatives: some useful formulae follow. sin? x + cos" = 1 tan? +1 = sec? 2 wee 1 cos(2z) es 1 | cos(2z) 2 2 ri 2 sin(22) = 2sinz cosa Improper integrals: look for 00 as one of the limits of integration; look for functions that have a vertical asymptote in the interval of integration Know the following limits. lime? = OO #490 Hint eo 8 = Note: this is the same as lim ¢* e450 zno0 dime Note: the answer is the same for lim L/:* and similar functions e450 300 lim 1/a = oO Note: the answer is the same for lim. 1/2” and similar functions a zl limInz= GO Fot08 lim nz = —OO 240 i = 1 jim arctan De