Logarithmic and Trigonometric Function Integration Examples

Ex4 die (x' In sx) Ex5 ( In /cosx1] = Cos x . ( Cos(n) = 3x' In(5x)+ 5x = cosx = 3x` (sx) + STY = - cosx = 3x' In(sx)+x' Ex 6 = d/e Ala [2 Inx + - = -x THE Ex 1 you Ing dy Iny = = Tia [Iny] = #/4 n(x-1) - - 3 ln(2*-1) 4 (2) = x-1 -

Ex 2 In 4/2 - siny + x ² == dia [in 2x-Iny - smy+x]= (0) 1/2 cosy.y' + 2X so cosy . y' + 2x=0 Intergral: du = mlul+c u= 5x-2 Ex1 S 5x-2 I dx du=Sdx S 1/4 dz/d Sudu 1/1/1/16 = 3 m Ex2 S 3x Inx dx 3 du Su+du u=Inx 11" - = xdu=dx = i (1xx) 1/h +

S Sin x dx cos x + c = - S tan u du = In /secu/tc S COS x dx = sin + + c S cot u du = In smultc S sec'x dx = tan x + c S sec u du = In/secution S sectan x dx = sec x+c S csc u do = In /cscu-cotu/+c - Scsex cotx dx= - csc x tc S csc'xdx = -cotx tc I Ex Sx sec(x) dx u = x' Ex S Sec x dx I du= 2xdx = Sx sec u du ==dx : S sec x. see Seex + tanx tans dx = I 1/2 2 See u du = S sec' Seck tany , dr secx + tanx = 1/2 In/secu + tan al tc secrites = S Sec x dx du= (secr tanis see's) d* = 1/2 In |secx' t anx'/te S 1/4 du = In/u/+c = Ex S sin 2x u=1+sin** sinx dx du = 2 Sink cosx dx Ex S tanx dx u= cosx = Sink dx du - sinx dx S 2 sinx cosx dx Cosx sin'x du =dx = Sink du u -sim = In 1u/+c = - su du = In / 1+ sin'x / tc = -In/ul+c = -In I cos +C seex : In loosal" tc

6.1 Natural log Function expand Review log,**4--2*- Ex 1. In is 1. In 1 = 0 = 2. lnx+lny = - 3. (n g = lax-lay = 4. Ex2. in 5x Graph f(x) = in(x'+1)- www they are g(a) inverse = (n(x'11) - - 1/Inx g(a) Ex3. f flx) = In x' sin'( TTX) - - In', x + + 2 = In x 5 + - In (x'+2) & =5lnx Ex5 Sx'dx = 5(nxt 2 = = + c Ex4. Exb S 4x - xx = 3 [Inxs = 3 x5. 4 x + +c = in(*) 3x'+c