# INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

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1 Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7.. If two functions diffr y constnt, thy hv th sm drivtiv. 7.. Gomtriclly, th sttmnt f ( ) d F () + C y (sy) rprsnts fmily of curvs. Th diffrnt vlus of C corrspond to diffrnt mmrs of this fmily nd ths mmrs cn otind y shifting ny on of th curvs prlll to itslf. Furthr, th tngnts to th curvs t th points of intrsction of lin with th curvs r prlll. 7.. Som proprtis of indfinit intgrls (i) (ii) (iii) Th procss of diffrntition nd intgrtion r invrs of ch othr, d i.., f ( ) d f ( ) d nd f '( ) d f ( ) + C, whr C is ny ritrry constnt. Two indfinit intgrls with th sm drivtiv ld to th sm fmily of curvs nd so thy r quivlnt. So if f nd g r two functions such tht d d d f ( ) d g( ) d d, thn f ( ) d nd g ( ) d r quivlnt. Th intgrl of th sum of two functions quls th sum of th intgrls of th functions i.., ( f ( ) + g( ) ) d f ( ) d + g ( ) d.

2 MATHEMATICS (iv) (v) A constnt fctor my writtn ithr for or ftr th intgrl sign, i.., f( ) d f ( ) d, whr is constnt. Proprtis (iii) nd (iv) cn gnrlisd to finit numr of functions f, f,..., f n nd th rl numrs, k, k,..., k n giving ( kf ( ) + kf ( ) , kf n n( )) d k f( ) d + k f ( ) d kn fn( ) d 7..5 Mthods of intgrtion Thr r som mthods or tchniqus for finding th intgrl whr w cn not dirctly slct th ntidrivtiv of function f y rducing thm into stndrd forms. Som of ths mthods r sd on. Intgrtion y sustitution. Intgrtion using prtil frctions. Intgrtion y prts Dfinit intgrl Th dfinit intgrl is dnotd y f ( ) d, whr is th lowr limit of th intgrl nd is th uppr limit of th intgrl. Th dfinit intgrl is vlutd in th following two wys: (i) Th dfinit intgrl s th limit of th sum (ii) f ( ) d F() F(), if F is n ntidrivtiv of f () Th dfinit intgrl s th limit of th sum Th dfinit intgrl f ( ) d is th r oundd y th curv y f (), th ordints, nd th -is nd givn y f ( ) d ( ) lim f( ) + f ( + h) +... f ( + ( n ) h) n n

3 INTEGRALS 5 or f ( ) d lim h f( ) + f ( + h) f ( + ( n ) h), h whr h n s n Fundmntl Thorm of Clculus (i) Ar function : Th function A () dnots th r function nd is givn y A () f ( ) d. (ii) First Fundmntl Thorm of intgrl Clculus Lt f continuous function on th closd intrvl [, ] nd lt A () th r function. Thn A () f () for ll [, ]. (iii) Scond Fundmntl Thorm of Intgrl Clculus Lt f continuous function dfind on th closd intrvl [, ] nd F n ntidrivtiv of f. f ( ) d F( ) [ ] F() F() Som proprtis of Dfinit Intgrls P : f ( ) d f () t dt P : f ( ) d f ( ) d, in prticulr, f ( ) d P : f ( ) d f ( ) d + f ( ) d c c

4 6 MATHEMATICS P : f ( ) d f ( + ) d P : f ( ) d f ( ) d P 5 : f ( ) d f ( ) d + f ( ) d P 6 : f ( ) d f ( ) d,if f ( ) f ( ),, if f ( ) f ( ). P 7 : (i) f ( ) d f ( ) d, if f is n vn function i.., f ( ) f () (ii) f ( ) d, if f is n odd function i.., f ( ) f () 7. Solvd Empls Short Answr (S.A.) Empl Intgrt + c w.r.t. Solution + c d ( ) d d+ c d 5 9c C. 5

5 INTEGRALS 7 Empl Evlut c Solution Lt v + c, thn dv c d d Thrfor, d + c c dv v log c C c. Empl Vrify th following using th concpt of intgrtion s n ntidrivtiv. d log C Solution d log C d +. Thus + log + + C d + Empl Evlut d,. Solution Lt + d I d d + sin + I,

6 8 MATHEMATICS d whr I. Put t d t dt. Thrfor I dt t + C C Hnc I sin C. Empl 5 Evlut d ( α)( β ), β > α Solution Put α t. Thn nd d tdt. Now t t t I t tdt ( β α t ) ( β α t ) dt k dt t, whr k t sin C sin C k + α β α +. Empl 6 Evlut tn 8 sc d Solution I tn 8 sc d 8 tn sc sc d ( ) 8 ( + ) tn tn sc d

7 INTEGRALS 9 8 tn sc d+ tn sc d 9 tn tn + + C. 9 Empl 7 Find d + + Solution Put t. Thn d dt. Now I d t dt + + t + t+ t A B Considr + t + t+ t+ t+ Compring cofficint, w gt A, B. Thn I dt dt t+ t+ log t+ log t+ log + + C + d Empl 8 Find sin + 5cos Solution Dividing numrtor nd dnomintor y cos, w hv I sc d tn 5

8 5 MATHEMATICS Put tn t so tht sc d dt. Thn dt dt I t t + t tn + C 5 5 tn tn + C 5. Empl 9 Evlut 7 5 d s limit of sums. Solution Hr,, nd h +, i., nh nd f () 7 5. n Now, w hv ( 7 5) d lim h f ( ) + f ( + h) + f ( + h) f ( + ( n ) h) Not tht h f ( ) 7 5 f ( + h) 7 + 7h 5 + 7h f ( + (n ) h) 7 (n ) h. Thrfor, ( ) ( ) ( ) 7 5 d lim h + (7 h ) + ( h ) (7 n h ). h ( ) lim h 7h n n h

9 INTEGRALS 5 h ( n ) n 7 ( )( ) h lim h 7 h.n lim nh nh h nh Empl Evlut cot tn tn d Solution W hv I cot tn tn d...() 7 tn d y (P 7 7 ) cot + tn cot cot 7 ( ) d d + tn () Adding () nd (), w gt 7 7 tn + cot I 7 7 d tn + cot d which givs I.

10 5 MATHEMATICS Empl Find 8 d + Solution W hv I 8 d +...() 8 ( ) d y (P ) I 8 + d () Adding () nd (), w gt Hnc I 8 I d 8 6 Empl Find Solution W hv + sin d I + sin ( sin + cos ) d d ( sin + cos ) d

11 INTEGRALS 5 ( cos + sin ) I. Empl Find tn d. Solution I tn d d d + tn. tn d + tn + log + + C. 6 6 Empl Find + d Solution W hv I d d Put t, thn dt d. I t + Thrfor, ( ) dt t 9 9 t log t t 9 C 9 ( ) ( ) log ( ) + ( ) C.

12 5 MATHEMATICS Long Answr (L.A.) Empl 5 Evlut d. + Solution Lt t. Thn t t A B + + t + t ( t+ ) ( t ) t+ t So t A (t ) + B (t + ) Compring cofficints, w gt A, B. So Thrfor, d d d tn + log + C 6 + Empl6 Evlut d 9 Solution W hv I d 9 d d 9 9 I + I.

13 INTEGRALS 55 Now Put I 9 t 9 so tht d dt. Thrfor I dt t log t C log 9 C + Agin, d I 9. Put u so tht d du. Thn du u I u log C 6 u + log + C. Thus I I + I + log 9 log + C +. Empl 7 Show tht sin log ( ) + sin + cos Solution W hv I sin d sin + cos

14 56 MATHEMATICS sin d sin cos (y P) + I cos d sin + cos Thus, w gt I d cos sc d log sc tn + log sc tn log sc tn + + log ( + ) log( ) + log ( ) log + log ( + ) log +. Hnc I ( ) Empl 8 Find ( tn ) d

15 INTEGRALS 57 Solution I ( tn ) d. Intgrting y prts, w hv I ( tn ) tn. d +.tn + d I, whr I + tn d Now I + tn + d tn d tn + d I ( ( ) ) tn I Hr I tn d ( tn ) + d ( log ) + log. Thus I log

16 58 MATHEMATICS Thrfor, I log + log 6 log 6 +. Empl 9 Evlut f ( ) d, whr f () , if < Solution W cn rdfin f s f ( ) +, if <, if < (y P ) Thrfor, f ( ) d ( ) d + ( + ) d + d Ojctiv Typ Qustions Choos th corrct nswr from th givn four options in ch of th Empls from to. Empl ( cos sin ) dis qul to (A) cos + C (B) sin + C (C) cos + C (D) sin + C

17 INTEGRALS 59 ' Solution (A) is th corrct nswr sinc f ( ) + f ( ) d f ( ) + C. Hr f () cos, f () sin. d Empl is qul to sin cos (A) tn + cot + C (C) tn cot + C (B) (tn + cot) + C (D) (tn cot) + C Solution (C) is th corrct nswr, sinc d I sin cos ( sin + cos ) sin cos d sc d + cosc d tn cot + C Empl If d + log C, thn (A) (C) 7, (B) 8 8 7, (D) 8 8 7, 8 8 7, 8 8 Solution (C) is th corrct nswr, sinc diffrntiting oth sids, w hv ( 5 ) + + 5, giving 5 ( + 5 ) + ( 5 ). Compring cofficints on oth 7 sids, w gt + nd This vrifis,. 8 8

18 6 MATHEMATICS + c Empl f ( ) d + c is qul to (B) f ( + c) (A) f ( c) d d (C) f ( ) d (D) f ( ) d Solution (B) is th corrct nswr, sinc y putting t + c, w gt I f ( c+ t) dt f ( + c) d. Empl If f nd g r continuous functions in [, ] stisfying f () f ( ) c c nd g () + g ( ), thn f ( ). g( ) d is qul to (A) (B) f ( ) d (C) f ( ) d (D) f ( ) d Solution B is th corrct nswr. Sinc I f ( ). g( ) d ( ) f ( ) g( ) d f ( ) g( ) d f ( ) d f ( ). g( ) d f ( ) d I

19 INTEGRALS 6 or I f ( ) d. Empl 5 If + 9t y dt nd d y d y, thn is qul to (A) (B) 6 (C) 9 (D) Solution (C) is th corrct nswr, sinc dt + 9t d dy + 9y y which givs 8y d y d + 9y. dy d 9y. Empl d is qul to + + (A) log (B) log (C) log (D) log Solution (B) is th corrct nswr, sinc I + + d d ( + ) d [odd function + vn function] + d d + ( + ) log + log.

20 6 MATHEMATICS Empl 7 If t dt, thn + t ( + t) t dt is qul to (A) + (B) + (C) (D) + + Solution (B) is th corrct nswr, sinc I t + t dt t t + + t ( + t) dt (givn) Thrfor, t +. ( + t) Empl 8 cos d is qul to (A) 8 (B) (C) (D) Solution (A) is th corrct nswr, sinc I cos d cos d cos d + cos d + cos d 8. Fill in th lnks in ch of th Empls 9 to. Empl 9 sin cos 6 8 d.

21 INTEGRALS 6 Solution 7 tn 7 + C Empl f ( ) d if f is n function. Solution Odd. Empl f ( ) d f ( ) d, if f ( ). Solution f (). n sin d Empl n sin + cos n. Solution. 7. EXERCISE Short Answr (S.A.) Vrify th following :. d log ( + ) + C + +. d log + + C + Evlut th following:. ( + ) d. + 6log log 5log log d

22 6 MATHEMATICS ( + cos) d 6. + sin tn sc d 8. d + cos sin + cos d + sin 9. + sind. d (Hint : Put + z). +. d (Hint : Put z ). + + d d dt t t 6. d d 8. d 9. d put t. d. sin d ( ). ( cos5 + cos ) d. cos 6 6 sin + cos d sin cos

23 INTEGRALS 65. d 5. cos cos d cos 6. d (Hint : Put sc θ) Evlut th following s limit of sums: 7. ( + ) d 8. d Evlut th following: 9. d. + tn d tn + m. d ( ). ( ) d +. sin cos d. d ( + ) (Hint: lt sinθ) Long Answr (L.A.) 5. d 6. d ( )( ) 7. sin 8. + d ( )( + )( )

24 66 MATHEMATICS tn d. sin d + (Hint: Put tn θ). + cos 5. ( cos ) cos d. tn d (Hint: Put tn t ). d ( cos + sin ) (Hint: Divid Numrtor nd Dnomintor y cos ) 5. log (+ d ) 6. log sin d 7. log (sin + cos d ) Ojctiv Typ Qustions Choos th corrct option from givn four options in ch of th Erciss from 8 to cos cosθ d is qul to cos cosθ (A) (sin + cosθ) + C (C) (sin + cosθ) + C (B) (sin cosθ) + C (D) (sin cosθ) + C

25 INTEGRALS sin d sin is qul to (A) sin ( ) log sin( ) sin( ) sin( ) + C (B) cosc ( ) log sin( ) + C (C) cosc ( ) log sin( ) sin( ) sin( ) + C (D) sin ( ) log sin( ) + C 5. tn d is qul to (A) ( + ) tn + C (B) tn + C (C) tn + C (D) ( + ) tn + C 5. d is qul to + (A) C + + (B) C + + (C) ( ) + + C (D) ( ) + + C 5. 9 ( + ) 6 d is qul to (A) C (B) C (C) ( + ) 5 + C (D) C

26 68 MATHEMATICS 5. If d ( log + + )( + ) + tn + log + + C, thn 5 (A), 5 (C), 5 (B), 5 (D), 5 5. is qul to + (A) (C) + + log + C (B) log+ + C (D) + log + C + log+ + C sin d is qul to + cos (A) log + cos + C (B) log + sin + C (C) tn + C (D).tn + C 56. If d ( ) C, thn (A), (B), (C), (D),

27 INTEGRALS d is qul to +cos (A) (B) (C) (D) 58. sind is qul to (A) (B) ( + ) (C) (D) ( ) cos sin dis qul to. + d ( + ). Fill in th lnks in ch of th following Ercis 6 to If d, thn. + 8 sin 6. d. + cos 6. Th vlu of sin cos d is.

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