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  1. MCQ’S OF INTEGRATION Anti-derivative: Let f(x) and F(x) be two functions of x such that ? ??(?(?)) = ?(?), then F(x) is called an anti-derivative of f(x) or an integral of f(x) with respect to x and written as, ∫?(?)?? =?(?) + ? , Where C is an arbitrary constant called constant of integration. Integrals of Some Elementary Functions: 1.∫???? =??+? ?+?+ ?; ? ≠ −? ; ∫(?? + ?)??? = (??+?)?+? 2.∫?.?? =? + ? ; ∫?.?? =?∫?.?? = ?? + ? 3.∫ + ?; ? ≠ −? ?.(?+?) ? ??? = ??|?| + ? ; ∫ 4.∫???? =??+ ? ; ∫???+? ?? =???+? ??+??? = ??|??+?| ? + ? ? + ? ? ?? ? √??? = ?√? + ? ?′(?) √?(?)?? = ?√?(?) + ? 5.∫???? = ??|?| + ? ; ∫ ?′(?) ?(?)?? = ??|?(?)| + ? ; ∫ 6.∫ Integrals of Some Trigonometric Functions 1.∫??? ?? = − ???? + ? 2. ∫??? ?? =???? + ? ?. ∫????? ?? =???? + ? 4. ∫??????? ?? = − ???? + ? 5.∫???? .???? ?? =???? + ? 6. ∫????? .???? ?? = − ?????? + ? Rules of integration: 1. Coefficent rule: ∫?.?(?) ?? =?∫?(?)?? 2. Sum/ difference rule: ∫[?(?) ± ?(?)] ?? =∫?(?)?? ± ∫?(?)?? 3. Integration by parts: ∫?.? ?? =?∫?.?? − ∫[? ??(?).∫? ?? ]?? Practice Questions: 1.∫(?2− 3? + 4)7(2? − 3)?? = (?2−3?+4)8 a.a. 7(?2− 3? + 4)6 c. ??(?2− 3? + 4) d. b. none 8 2.∫{?.???? + ????}?? = a.a. ?.???? 3.∫??{???? + ??????}?? = b. ?.???? c. ?.????.???? d. ?????.????.

  2. MCQ’S OF INTEGRATION a.a. ??(??????) ???? ???????? = a.a. (??????) ?tan−1? 1+?2?? = ??(??????) ?????? b. c. d. ?????? 4.∫ b. (???????) c. ??(??????) d. ??(??(????)) 5.∫ ?tan−1? tan−1(?tan−1?) ?? ?tan−1? a. ??(1 + ?2) b. c. d. ?sin−1? √1−?2 ?? = a.a. ??(1 − ?2) 1 1−?? ?? = 6.∫ ?sin−1? sin−1(?sin−1?) ?? ?sin−1? b. c. d. 7.∫ 1−?? 2 ?? (1 − ??) ??(1 − ?−?) a. √1 − ?? b. c. d. 8.If ?′(?) = ?2+ 2 and ?(0) = 0, then ?(?) = 3 2 a.a. ?3 ? 2+?2 ?2 2+ 2? + 5 + ? 2?2+ 2? + ? 3+ 2? b. c. 4+ ? d. ???2? ?????? = 9.∫ a. 2???? + ??(???? + ????) + ? b. 2???? − ??(???? + ????) + ? c. 2???? + ??(???? + ????) + ? d. 2???? − ??|????| + ? ???2?_???2? ????−?????? = 10.∫ a. 2???? + 2????? b.???? − 2???? + ? c.???? − 2????? d.???? − ???? 1 11.∫ √?+1−√? ?? = 3 2+? 3 2 3 2+ ? 3 2] b.(?+1) a. 2 2 √?+1−√? 3 d.2 3 [2√? + 1 + 2√?] c. 3 [ (? + 1) 3 12.∫cot23??? = −?????3? −??????.???? −???3? a.−???3? b. c. d. − ? 3 3 3 13.∫??(sin2? + ???2?)?? = a.??sin2? 14.∫ a.? + 4 ??(? − 2) b. ?2?−1 ??−1 ?? = a.??− ? ?????2? ????2? ????2.???? b. c. d. ?+2 ?−2 ?? = ?????2? ????2? ????2?.???? c. d. 15.∫ ?? 2− ? (??−1)2 ?2? 2+?2 b. c. d. 2 2

  3. MCQ’S OF INTEGRATION ?? 16.∫ ??+1?? = a. 1 ?? ??b. ??(1 + ??) ?? c. 1+?? d. 17.∫{sin−1(tan ???−1?) + cos−1(tancos−1?)}?? = (sin−1?)2 ? 2 ?? 2 a. 1 b. c. d. 2 18.∫555?.55?.5??? = 555? (?? 5)3 19.∫ 555?.(?? 5)3 555?.(?? 5) 555?.55?.(?? 5)3 a. b. c. d. 1 √2??−?2 ?? = −(?−?)2 ???−1(?−? b. tan−1(?−? √?−? 2 a. ?) ?) c. d. 2 20.∫?????.????.???2??? = ?−???2? ?????.???? ?????.???? ????2? a. b. c. d. 2 2 sin4? cos6? ?? = 1 3tan3? 22.∫?? ?? ?.???? = 21.∫ 5 2? (????.????)5 1 5tan5? a. b. c. d. sec 5 (??)? 1+?? ? ?? a. ???? 23.∫(???? + ????2?)?? = a. 2 24.∫??(??? +1 ?? b. c. ?? ?? d. ????.???? ????? ?.sec2? b. c. ?.???? d. 2 ?)?? = ?? ?? ? ?? ? ?.?? ??.?? ? a. b. c. d. 25.If ?? ??= (1 − √?)(1 + √?), then ? = a. 26.∫(2sin 2? +1 ? −?2 ? 2− 2?2+ ? ?2+ ? b. c. 2+ ? d. −1 + ? ?)?? = a. ???2? + ?? ? b. ???2? + ?? ? c. −???2? − ?? ? d. −1 + ? 27.∫(??? ?6)?? = ?6 6 ?7 7 ?6 ?? ? ??6 a. b. c. d. 28.∫????(sec? + ???? ) ?? = a. ????(sec? + tan ?) ?? = c. tan ? + sec? 1+cos2? sin2? ?? = a. -cot x- 2x 30.∫(1 + ? +?2 b. 2???? − tan? d. 2tan? − sec? 29.∫ b. −2cot? − 2? c. −2cot? − ? d. −2cot? + ? 2!+?3 3!+ ⋯?? ∞)?? = 1 + 2? + 3?3+ ⋯ b. ?? ??− 1 a. c. d. log?

  4. MCQ’S OF INTEGRATION 31.∫sec2?.cos(????)?? = a. sin (????) 32.∫(?.?2?)?? = ?2? 2+?2? 33.∫ √1−?2 (sin−1?)?2 b. sin (????) c. ?????(????) d. sec (????) ?2? 2−?2? ?2?(?2 2−?4 2.?2?− ?4? a. 4 b. 4 c. 4) d. (?????2(sin−1?)) ?? = −2 −cot(sin−1?) 2?????(sin−1?) a. b. √1−?2 c. d. 2 34.∫sec2?.?????+1) ?? = a. ????? (sec2?) √???? ?? = a. 2sec?.tan ? 36.∫ ?????+1 ????2? sec?.????? b. c. d. 35.∫ b. 2sec? c. 2√???? d. 2√???? 1+???? ?+??sec? ?? = a. ?? (??sec?) 37.∫|?|3?? = ?4 4 38.∫(1 ?? (sec?) 1+???? b. ?? (? + ????) c. d. ?? (? + ??sec?) −?4 3|?|2 a. b. 4 c. d. none ?sin( ?? ?)) ?? = sin(?? ?) 3?2 ?6+1 ?? = a. ?? (?6+ 1) a. b. −sin (?? ?) c. cos(?? ?) d. −cos(?? ?) 39.∫ 3tan−1(?3 tan−1(?3) 3tan−1(?3) b. c. d. 3) 40.∫??((1 1 ?2))?? = ?− −?? ?? ?2 ?? ? −?? a. ?2 b. c. d. ?

  5. MCQ’S OF INTEGRATION 41. ∫2√1+???2? (???? − ????)?? ?? 2√1+???2? a. 2√1+???2?.???2+ ? b. + ? ???2 2√1+???2? c. 2√1+???2?..???4+? d. + ? ???4 42. ∫[sin(????)+ cos(????)?? ] is equal to a. ????(????) + ? b. ????(????)+ ? c. ??sin(????)+ ? d. ??cos(????)+ ? Integration of ∫?3tan−1?4 43. ?? is 1+?8 (tan−1?4)5 (tan−1?4)2 (tan−1?4)2 (tan−1?4)2 a. + c b. + ? c. + ? d. + ? 10 2 8 4 Integration of ∫234?34?4? dx is equal to 44. 234? ???12+ ? d. 234?.???2.???3.???4+ ? 234? 234?.???24+ ? a. b. ???2.???3.???4+ ? c. If ? = tan−1√1+???3? 1−???3? then ∫??? is equal to 45. ? 4 (? + 3?) ? 4(? + 3?) ? 4 (3? + ?) ? 4(3? + ?) a. b. c. d. 46. Integration of ∫???? ???2? ???3? ?? is sec22? sec23? sec22? −sec23? sec2? + b. sec2? = a. + ? + ? + 2 3 2 3 ??????? +1 1 3??????3? d. −??????? −1 2??????2? +1 c. 3 ??????3? + ? 2??????2? + ?? 1+?? is equal to 47. Integration of ∫ 1 log(1 + ??) + ? log?.log(1 + ??) + ? a. b. 1 log(1 − ??) + ? ??? ?log(1 − ??) + ? c. d. ?? 48. Integration of ∫ ????(????)cot (????) is equal to a. sin(????) + ? b. cos(????) + ? c. tan(????) + ? d. sec(????) + ? cot2√? √? 49. Integration of ∫ ?? is equal to a. −2?????√? − 2√? + ? b. −2???√? − 2√? + ?

  6. MCQ’S OF INTEGRATION c. 2???√? + 2√? + ? d. −2???√? + 2√? + ? Integration of∫(1 − ???? + cos2? − cos3? + ⋯) ??,???? ≠ 1 is 50. 2tan? tan? 4tan? 1 2tan? a. 2+ ? b. 2+ ? c. 2+ ? d. 2+ ? ?3?? √4−?8 is 51. Integration of ∫ 1 4sin−1(?4 1 4sin−1(?4 1 4sin−1(?4 1 4sin−1?4+ ? a. 4) + ? b. 3) + ? c. 2) + ? d. 52. Integration of ∫sin(????) ?? a. ?[sin(????) − cos(????)] + ? b. ?[sin(????) + cos(????)] + ? ? [sin(????)−cos (????) c. ] d. sin(????) − cos (????) 2 53. Integration of ∫???√??? a. −2√????√? − 2???√? b. −2√????√? + 2???√? c. 2√? cos√? + 2???√? d. 2√????√?_2???√? Integration of ∫?√? ?? 54. 2√??√?+ 2?√?+ ? b. 2√??√?− 2??+ ? a. 2√??√?− ?√?+ ? d. √??√?− 2 ?√?+ c. 1 ??−1?? =: 55. ∫ d) ??−1 b)log(??+ 1)+ c c) log(1− ?−?)+ c a) √??−1 + ? +? 2 56. ∫(?+?)(?+????)? ?? = ? b) (?+????)4 c) (?+????)4 d) (?+????)4 a) (? + ????)4+ ? + ? + ? + ? 4? 4 ? x sin ? x cosx−????−1?? = : 57. ∫ |????? − ???? − 1| + ? |????? −???? −1| +? a) −log b) log |????? −???? −1| +? |????? − ???? − 1| + ? c) −log d) log `

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