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Quiz 4 Information Math 208 page 1 Quiz 4 will cover the following material: (all handouts posted on the web site so far) 1. All material for Quizzes 1-3 2. Di¤erentiate any function, including logarithmic, exponential, and inverse trigonometric functions. 3. Apply the fundamental theorem to compute de…nite integrals and di¤erentiate functions de…ned using de…nite integrals. 4. Graph trigonometric functions (all 12 of them) and state their basic propoerties, di¤erentiate them and integrate them. (Exception: we did not yet integrate sec 1 x and csc 1 x; so you don’t need to know those.) 5. Integrate using substitution, trigonometric substitution, integration by parts, and partial fractions with linear factors. Integrate trigonometric and inverse trigonometric functions. 6. Approximate logarithmic expressions using the de…nition of ln x as Zx 1 dt. t 1 Sample Quiz 4 dy 1. Compute if y = dx sec Zx 1 p 4 1 dt 1 + e2t 2. Compute each of the following integrals. a) b) c) d) e) f) g) h) Z Z Z Z Z Z Z Z 1 (x + 1) (x dx 1) x (x + 1) (x 1) dx i) Z p j) Z2 p x x 1) dx Z3 k) 5 dx 2 x +x sin 2x dx 1 + cos 2x l) m) Z n) Z o) Z2 sec3 xdx dx Z 0 p) Z p 1 + cos xdx 0 1dx x2 q) Z=4 r) Z 1 dx arctan x dx 1 + x2 s) x2 cos (2x) dx t) p p 1 9x2 1 4 tan2 xdx 0 1 2 1 dx 2 x + 2x + 2 9 9 dx 1 x2 (x + 1) (x 1 p 4x2 x 4x2 x2 25 dx dx u) v) Z Z Z Z sec3 x tan3 xdx ex dx ex + 1 xe x dx sin 2x sin 10x dx x4 ln x dx Last revised: February 9, 2017 Quiz 4 Information Math 208 page 2 Answers sec x tan x 1. p 4 1 + e2 sec x 2. a) 1 ln jx 2 d) 5 ln jxj g) j) n) 1j 1 ln jx + 1j + C 2 5 ln jx + 1j + C b) 1 ln x2 2 e) arctan jx + 1j + C 1 (sec x tan x + ln jsec x + tan xj) + C 2 16 15 k) 1 1 ln 2 + ln 3 2 2 p 1 ln 18x + 6 9x2 3 s) ln (ex + 1) + C e x o) xe h) x 2 +C c) x + p 1 ln 2x + 4x2 2 m) p p) 2 2 u) 1 sin 8x 16 1 ln jx 2 1 ln jx + 1j + C 2 1j 1 ln jcos 2x + 1j + C 2 f) 1 arctan2 x + C 2 l) 25 + C t) 1 +C 9 +C 1 x cos 2x 2 q) 1 i) 1p 2 4x 4 9+C 1 1 sin 2x + x2 sin 2x + C 4 2 r) 4 1 sin 12x + C 24 1 sec5 x 5 v) 1 sec3 x + C 3 1 5 x ln x 5 1 5 x +C 25 Last revised: February 9, 2017