Question

Question 4 a) Differentiate with respect to x, i. y = sin 2x ii. y = x In(5x + 2) b) Show that if y = cotx, dy dx -cosec? x c) Show that if y = tan x, then dy dx 1 1+xal Question 5 Use calculus to find any turning points of the function A(t) = te-020 and determine their nature (maximum, minimum or inflexion) using any method. Question 6 a) Find tan” x dx b) Use integration by parts to find In x dx c) Use the substitution u = 1 + e* to show that Sites de dx = 5e2 - e* + In(1+e*)+c Use any relevant trigonometric identities to simplify the integrand and evaluate the following integral. sinx 1 + cosx

Answer 1

(4) (2) (i) e} pod Sin 200 Sin (200) 30) dur (Sin 2x) Sin ² (200) cosan 200) 30 (3 Sin (20) 3x Sink 20 alij y - Uc ع عن 3 dn (5x+2) oc de la 5x+2) de cord). (536+2) 3 x2 In (SJC+2) [5x + 3 (5812) de 15x+2) х 43 5x+2 Oc? (5x+2) g 11 y = cotx d con (cony y' = Sinse Sinx du cocinay Senzor y Sin²x Colx sin²n Cosecx note & (Sinlue + cos²) (Six2n) sinan oc ОС © y = tanto Secay. du i = 1 dy tay Casey Stery= uc T Boty aboy Cosy = doc Hoc2 Vuc²ti (5 -ont +2t eoret 0 2t 2 O2 Ald) = te-02 Alt) - tex x -0.2 e it [v.2t + 2] =0 - = 0, - oist + a=0 A" (t) = & Ceoist (at - orat? ( 2-0-44) - 0.2.202t (et - 0.24) A" (0) = 2 70 to All (10) = 2 (2-4) to local maximum. = lo 80.24 pain of 0.22 (20-20) Docal minimum -2 ez

is beind of minimum to + - masina. is farkut of tot A"(t) = é est [ 2-ooft - ootut + 0.0t th] 2 2 2 Li- of 02 t? -0.44 +=0 11 -0.44 +0.0282] 10o f² 27 +1=0 ? - 20t +50=0 L. a x² + b 2 + (no ich b² - 4ac to then imaginary rest b²-4ac7p the real root. red rost 20 + r 200 101552 - 200 to 6² 40C = 400 11 20+ lo 52 2 - t - lod sia are (- - 553) (4-(40+502) inflection point. lo-552) concame up becaus 600 ton Alt ei ACHI > 0 Allt) as (loose , lot SE) lot 5821 Concare down ANH Concare up. 6 la) s tan? uc dx = (Sec? or -1) do teu x - x + c 1 ton x =deczon) son classkon Let u = lnur = 1 continued continued Sdn Uc3 202

2-2e" thellte I to, u ordo 0 Per coulden ur - 1 da - (la sc) २० 2002 fc post shot) + 2 -dot c 03 | cho tc 2002 40CP - [(ale x) + 1] + c 4c? S 300 e 1+ don it 30t - 3x A-1 eur Let evol = (U- U= Ite durcle) = U- 1 = du = ex du doc du ex = ) du H eur dor ( suy 3. du (a) е (ay S sayech S (W²-2uti da +4) dy allt lan & C = (ltet) २ altex) + ln (Hetc, - e tl (tex) + + e 2 2 ten - (C-3 11 - exth Lite te c= 6-2