BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I
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1 EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si )/ (cos ) / d Q. (si )(sec ) Q. d Q.6 si d Q.7 l l d Q. (l ) (l ) l d Q.9 e d Q. Itegrte f () w.r.t., where f () = t + l l ( )d Q. Q. ( ) si d Q. (e cos ) d cosec cot. sec Q. cosec cot sec d Q.7 sisec Q. d ( cos si )( si cos ) cossi d d Q. d Q.6 79si d seccos ec Q. t.t.t d Q.9 si si si cos Q. d Q. si cos cos cos d si d Q. e Q.6 si d Q. si t si cos ( si si cos Q.9 / (7 ) cos ) d Q.7 d ( b) d c ( b) l d Q. / d Q. d ( ) Q. Q. e ( ) d d Q. d Q. cot t si 7 d Q. ( ) 7 d
2 d Q. Q. Q. d 6 d 9cos si Q. Q.6 cos Q.9 si z si si EXERCISE II / / Q. si rc t(si) d Q. cos. si 6 d Q.7 Let h () = (fog) () + K where K is y costt. If h() d d d Q.7 ( ) ( )( ) ( )d d Q. cos e Q. Evlute I = / d Q.6 cos (cos si ) f () f (t) vlue of j () where j () = dt, where f d g re trigoometric fuctios. g(t) g() d d = (, ) (l ) d hece fid I. si the compute the cos (cos ) Q. Fid the vlue of the defiite itegrl si cos d. Q.9 Evlute the itegrl : d Q. If P = d ; Q = d d R = () Q =, (b) P = R, (c) P Q + R = Q. Prove tht eb g d zb ( )( b) b d b ( ) ( b) ( b) j the prove tht Q. ( ) d Q..l d Q. Evlute: d Q. si d Q.6 / si si bcos d Q.7 d si Q. e cos d Q d Q. Let, be the distict positive roots of the equtio t = the evlute (si si ) d, idepedet of d.
3 / Q. cos si si Q. Evlute: d Q. ( )si d ( cos ) ( b)sec t d (,b>) t Q. If, d re the three vlues of which stisfy the equtio cos ) (si d cos d = the fid the vlue of ( Q. Show tht pq ). cos d = q + sip where q N & p Q.6 Show tht the sum of the two itegrls / e (+)² d + e 9( /)² d is zero. / si Q.7 d / Q. si si b cos d (>, b>) Q.9 d Q. / t si si b si d Q. si b Q. Commet upo the ture of roots of the qudrtic equtio + = k + t k dt depedig o the vlue of k + R. Q. si z z Q. Show tht e e dz e e d Q. Prove tht z dz Q.6 (.d ) (b d d si.si.cos d ) / ( > ) d Q.7 ()., (b) d l L si d d Q. Show tht = = cos cos NM Q.9 si si d si if if (, ) (, )
4 Q. Q. (sicos) d ( si )cos d ( cos ) Q. Evlute l Q. Prove tht u f ( t) dt du = d Q. f (u).( u) du. 6 t d Q. Lim (6si 7 cos ) d. Q.6 Show tht f d f ( ). l l. ( ). ( Q.7 Evlute the defiite itegrl, Q. Prove tht () ( )( ) d = (b) 66 si 69 ) d d = d (c) d ()( ) = where, > (d).d ()( ) = where < Q.9 If f() = cos (cos ) (cos ) (cos ) (cos ) (cos ) cos, fid f( ) d Q. Evlute : e l t si (cos ) d. Q. If the derivtive of f() wrt is Q. Fid the rge of the fuctio, f() = EXERCISE III cos f () the show tht f() is periodic fuctio. si dt t cos t. Q. A fuctio f is defied i [, ] s f() = si cos ; ; f() = ; f (/) =. Discuss the cotiuity d derivbility of f t =. Q. Let f() = [ if if d g() = cotiuity d differetibility of g() i (, ). Q. Prove the iequlities: () 6 < d f(t) dt. Defie g () s fuctio of d test the (b) e / < e d < e².
5 (c) < d b the fid & b. cos d (d) Q.6 Determie positive iteger, such tht e ( ) d = 6 6e. Q.7 Usig clculus () If < the fid the sum of the series.... (b) If < prove tht (c) Prove the idetity f ()= t + t + t Q. If () = cos ( t) (t) dt. The fid the vlue of () + (). 7 d y Q.9 If y = f (t) si ( t) dt the prove tht y = f (). d Q. If y = ltdt, fid d y d t = e t = cot Q. If f() = + [y² + ²y] f(y) dy where d y re idepedet vrible. Fid f(). 7 cot dy Q. A curve C is defied by: = e cos for [, ] d psses through the origi. Prove tht the d roots of the fuctio (other th zero) occurs i the rges < < d < <. Q.() Let g() = c. e & let f() = e t. ( t + ) / dt. For certi vlue of 'c', the limit of s is fiite d o zero. Determie the vlue of 'c' d the limit. (b) Fid the costts '' ( > ) d 'b' such tht, Lim t d t t b si =. f( ) g( ) d t Q. Evlute: Lim dt d (t )(t ) si Q. Give tht U = {( )} & prove tht Q.6 If d U d = ( ) U ( )U, further if V = e. U d, prove tht whe, V + ( ).V ( ) V = t t dt = ( > ) the show tht there c be two itegrl vlues of stisfyig this
6 equtio. if Q.7 Let f() = if. Defie the fuctio F() = f(t) dt d show tht F is ( ) if cotiuous i [, ] d differetible i (, ). Q. Let f be ijective fuctio such tht f() f(y) + = f() + f(y) + f(y) for ll o egtive rel & y with f () = & f () = f(). Fid f() & show tht, f() d (f() + ) is costt. Q.9 Evlute: () Lim (b) Lim (d) Give Lim /... ; C C... ; (c) Lim Q. Prove tht si + si + si si (k ) = prove tht, / Q. If U = si si / si k d = si! = b where d b re reltively prime, fid the vlue of ( + b) k / ; si k, k N d hece si d, the show tht U, U, U,..., U costitute AP. Hece or otherwise fid the vlue of U. Q. Solve the equtio for y s fuctio of, stisfyig y (t)dt ( ) t y(t)dt, where >, give y () =. Q. Prove tht : () I m, = m. ( ) d = (b) I m, = m!! ( m )!! m. (l ) d = () (m ) m, N. m, N. Q. Fid positive rel vlued cotiuously differetible fuctios f o the rel lie such tht for ll f () = ( t) f '(t) f dt + e Q. Let f() be cotiuously differetible fuctio the prove tht, where [. ] deotes the gretest iteger fuctio d >. [] [t] f (t) dt = []. f() f (k) Q.6 Let f be fuctio such tht f(u) f(v) u v for ll rel u & v i itervl [, b]. The: (i) Prove tht f is cotiuous t ech poit of [, b]. k
7 b (ii) Assume tht f is itegrble o [, b]. Prove tht, f( ) d ( b ) f( c) ( b ) Q.7 Let F () = deotes the derivtive., where c b t dt d G () = t dt the compute the vlue of (FG)' () where dsh Q. Show tht for cotiuously thrice differetible fuctio f() Q.9 Prove tht f() f() = f() + k ( ) k k f( ). = k m + f ( t )( t ) dt m k ( ) k m k k Q. Let f d g be fuctio tht re differetible for ll rel umbers d tht hve the followig properties: (i) f ' () = f () g () (ii) g ' () = g () f () (iii) f () = (iv) g () = () Prove tht f () + g () = 6 for ll. (b) Fid f () d g (). ANSWER KEY EXERCISE I Q. l cos cos + C Q. Q. l(cos + si ) + + (si os ) Q. t 6 l Q. t Q.6 e c e cos si cos si Q.7 (c) (si ) l t l (sec ) Q. l t Q.9 t t Q. l b b t t + t + C whe t = + / Q. / cos Q. cos. rc cos si. l cos si si si Q. t / (t ) Q. l t + sec² + t Q. rccos c
8 Q.6 ( + ) rc t Q.7 9. l Q. l (l) l e Q.9 l e e t t Q. l ( ) Q. t ( t ) t 6 l t + C where t = /6 Q. + t cos cos l Q. si sec (si cos ) (si cos ) Q. l c cos cos + c Q. C l( + ( + )e ) ( )e Q.6 si cos l t si cos Q.7 l rc t (si cos ) c Q. (sec ) (sec ) (sec ) si cos Q.9 si l si cos cot cot cot cot cot Q. l cos si Q. rct t c Q. cos si cot. l e cos cos Q. l ( + t) l ( + t ) + l t t t t t t where t = Q. lt t c Q. c ( ) cot Q.6 c e cos ( cosec ) Q.7 si b k Q. e c l Q. rcsec c Q. u u u Q. t t t si t u where t = t (7) Q.9 c 9 7 c where u Q. t si si cos Q. l t () + Q. rct c () Q.6 + C si Q.7. c Q. l + C
9 t t Q.9 l l t t where t = cos d = cosec (cot) Q. cosec t cosec Q. 6 EXERCISE II Q. l Q. 6 e Q. Q. 6 Q.7 sec() Q. 6 Q.9 + Q. Q.6 l 7 Q. ( l ) Q. l ( ) Q. Q.6 ( b) Q.7 Q. (e ) 6 Q.9 Q. ( b) Q. (b) Q. Q. ( ) Q. Q.9 Q. 6 rc t rc t Q. Q.7 Q. 6 Q. rel & distict k R Q. Q.6 Q.7 () ; (b) l Q.9 l Q. 6 l Q. 7 Q. l 6 Q. Q. 7 Q Q.9 Q. ( l) EXERCISE III Q., Q. cot. & der. t =
10 Q. g() is cot. i (, ); g() is der. t = & ot der. t =. Note tht ; ( ) for g() = for for Q. (c) = & b = 7 Q.6 = Q.7 () 6 Q. cos Q. + e Q. f() = + +9 ² 9 Q. () c = d Limit will be (b) = d b = Q.. Q.6 = or Q.7 F() = if if if Q. f () = + Q.9 () e (/) ( ) ; (b) l ; (c) e ; (d) Q. U = Q. y = e e Q. f () = e + Q.7 Q. f () = + e ; g () = e
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