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1 INTEGRATION of FUNCTION of ONE VARIABLE INDEFINITE INTEGRAL Fidig th idfiit itgrals Rductio to basic itgrals, usig th rul f ( ) f ( ) d =... ( ). ( )d. d. d ( ). d. d. d 7. d 8. d 9. d. d. d. d 9. d 9. ( ) d. ( ) d. si d 7. d si si 8. si d 9. ta d. ta d. cosh d. l si d. d. d l. (l ) / d. ( ) d 7. d si( ) (arcsi ) 8. d 9. d. d cos ( ). tah( ) d. sih () d. d. d. d Right or wrog?. si d = si. si d = cos. si d = cos si. ( ) ( ) d =. ( ) d = ( ). ( ) ( ) d =. d os. / 9 ( ). l. l.
2 l( ) l( ). arcta( ). 7. arcta( ) 8. arsh ( ) 9.. arcsi. arch ( ). 9 l( ) ( ) ( )... cos cot si cos cos 9. l cos. ta.. cos /. (l ) c. l l / (l ). 9. cos( ) ( ). (arcsi ). l( ) l sih(). l cosh( ).. sih ( ). arcsi. l. ta Right or wrog?. w. w. r. w. w. r Itgratio by parts f ( ) g( ) d = f ( ) g( ) f ( ) g ( ) d. ( ) d. cos( ) d. arctgd. ( ) l d. l( ) d. cos d cos( ) l( ). ( ). si( ). arctg 9. l. l( ) arctg (si os ). Itgratio of ratioal fuctios. d. d. d. d ( ) ( ) 7. d 8. d. d ( )( ). d ( )( ) 9. d
3 . l 8l. arcta l l l. l l l l( ). l l arcta 8 7. l 8. arcta 9. l Itgratio by substitutio. d. d. d. d 7. d. d si. d. d 8. d 9. d cos. d. d si cos. 8 arcsi 8. ar cosh( ) ( ) ( ).. ( ) / ( ) /. l( ). l( ) l( ) 7. l( ) 8.. c ta( / ). arcta ta 9. cot (arcsi ). cot( / ) DEFINITE INTEGRALS Eprss th limits as dfiit itgrals:. ck lim, whr P is a partitio of [,] P k =. k c P k = k k. lim, whr P is a partitio of [,]. ck k P k=. lim, whr P is a partitio of [,].
4 . d. d. d Fid th drivativ of. cos tdt. t dt. cos tdt si. dt t. dt t cos... cos.. Fid th rag valu of f ovr th giv itrval. At what poit or poits i th giv itrval dos th fuctio assum its rag valu?. ( ) =,. f ) si,. ( ) =,,, f, [ ] ( =, [ ]. f =, f()=. f =, ( ) = [,], f =, f()= f. [, ], = / Fid uppr ad lowr bouds for th valu of /. d. si( ) d. 8 d 7. d. d 7. Suppos that f is cotiuous ad that ( ) d= f, [ ] [ ] f, f ± / ) = f. os d (, f. Show that f ()= at last oc o [,] uppr boud=ub, lowr boud=lb. up=/, lb=/. ub=si, lb=. ub=, lb= 8. ub=, lb=. ub=/, lb= /. ub=, lb= 7. From th valu of itgral w gt f =. Th fuctio f is cotiuous, thrfor f taks o at last oc o th giv itrval.. Evaluat th itgrals:. ( ) d. d. ( ) d. d / 9. ( cos ) d. d
5 / 7. t ( t ) dt 8. t( t ) dt 9. d 9. ta d. d l. d. d / 8cosh 9. d 7. d 7. d. d l / si. d cos dt. 8 t l 8. sih d....-/.. / / 7. / 8. /8 9. ( ) /. l. (l-l). l. (l-l). /. /. / 7. / 8. (-l)/ 9. (sih-sih). ( ) APPLICATIONS of DEFINITE INTEGRALS Ara. Fid th total ara of th rgio btw th curv ad th X-ais a) y =, b) y =, c) y = /, 8. Fid th ara of th rgio closd by th curvs a) y =, y = b) y =, y = c) y =, y = d) = y, = y ) y =, y = f) y =, y =, = g) y =, y =, = h) y = /, y=, =. Fid th ara of th rgio boudd by th curvs a) y = /, y = / b) y =, y = c) = y, y / = d) y = ( ), y = ) y = si, y= / f) y = ( ), y =. Th rgio boudd blow by y= ad abov by y= is to b partitiod ito two subsctios of qual ara by cuttig across it with th horizotal li y=c. Fid th valu of c.. Dtrmi th ara of th rgio closd by th Y-ais, th graph of y = ad its tagt li touchig th curv at th poit whos abscissa is.. Fid th slop of th li y=m ( m positiv umbr), if th ara of th rgio closd by this li ad th graph of y= is qual to..a) 8/ b) 8 c) /.a) / b) 9/ c) 8/ d) 9/ ) / f) / g) / h) /-l.a) /8-l b) 9 c) 8/ d) / ) -/ f) /
6 . c = ( ). /. m= Volum. Rotat th giv curv about th X-ais ad dtrmi th volum of th gratd solid a) y = /, [,] b) y = cos, [ /, / ] c) y =, [,]. Rotat th giv curv about th Y-ais ad dtrmi th volum of th gratd solid a) y =, y [ /,] b) y= l, y [ /,/ ] c) y =, y [,]. Rotat th graph of th fuctio y = / ( y, a ) about both th Y-ais ad th X- ais. What is th valu of a if both solids h th sam volum?. a) b) ( ) c) ( ) c) (l l ). a=/9.a) (l ) b) ( ) Arc lgth. Calculat th arc lgth of th curv a) y =, b) y = /, y / c) =, y d) y =, 9 8 y. Fid th valu of b kowig that th arc lgth of th curv sgmt giv by th graph of th fuctio f ( ) = / ad lyig btw th poits a= ad b is qual to uits. 9. Fid th arc lgth of a) y =, l l 8 b) y= cos tdt, / /.a) 7/ b) ( ) c) / d) /. b=7.a) / b) 7 IMPROPER INTEGRALS A) Evaluat th itgrals:. d. 7.. d d ( )( ) d. d 8... d /. d ( ) 9. d. l d d. d arcta. d. d. d /. ta d
7 B) Tst th itgrals for covrgc (itgratio, compariso tst) l d d d. / d... si d. l 7. d os 8. d C) Fid th valus of p for which ach itgral covrgs. d p (l ). d p (l ). d 9. d A).. /... / / l.. /. l. (-/) / l B). covrgt, th valu of th itgral is., d si is covrgt thrfor th origial itgral is covrgt., d covrgt thrfor th origial itgral is covrgt., d covrgt thrfor th origial itgral is covrgt. covrgt. divrgt,, d l is divrgt 7. divrgt 8. divrgt 9. covrgt C). covrgt if p<, divrgt if p. covrgt if p, divrgt if p is is 7
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