Definite and Indefinite Integration

K

K E

E Y

Y

C

C O

O N

N C

C E

E P

P T

T S

S

1

1 . . D ED E F IF I N IN I T IT I O NO N ::

If f &

If f & g g areare functions functions of x such that g'(x) = of x such that g'(x) = f(x) f(x) then  then the functhe function g tion g is calis called aled aPRIMITIVE PRIMITIVE OO RR ANTIDE

ANTIDERIVATIVE RIVATIVE OO R INTEGRALR INTEGRAL of f(x) of f(x) w.r.t. x w.r.t. x and iand is s wriwrittetten symbolically asn symbolically as

f ci

f ci

II  f(x)  f(x) dx = g(x) dx = g(x) ++ c c < <==>> —  — (g(x) + (g(x) + c) =c) = f(x), f(x), where  where c is c is called called thethe  constant of integration,  constant of integration, dx

dx 2.

2. STANDARD STANDARD RESULTS RESULTS ::

\ n + l \ n + l r (ax+b) r (ax+b)nn i) i) _JJ ((aax x + + b)°dx= b)°dx= , , \ \ ++ c nc n ** -- ll   a(n+l)   a(n+l) iii) f 

iii) f  eeax+bax+bdx dx = - = - eeax+bax+b + c + c aa

v)

v) f sif si n(an(a x+bx+b ) ) dx dx = = -— co-— co s(s( axax +b+b ) ) + + cc aa

vii)

vii) j j tan(ax tan(ax + + b) b) dx dx == —  —  InIn sec (ax + b) + c sec (ax + b) + c aa

ix) J

ix) J secsec22 (ax + b) dx = (ax + b) dx = — —  tan(ax + b) + ctan(ax + b) + c aa

xi) J

xi) J sec (axsec (ax + + b) b) .. tan (ax tan (ax + b) + b) dx = dx = — —   sec (ax  sec (ax + b) + + b) + c c aa

xii) J

xii) J  co  cosec sec (ax (ax + + b) b) . co. cot t (ax (ax + + b) b) dx dx = = cosec cosec (ax (ax + + b) b) + + cc aa

xiii)

xiii) JJ secx secx dxdx =I=I n (secx n (secx + + tanx) + tanx) + cc OR OR  InIn tan { tan { + + y y jj + + cc

xiv) J

xiv) J cosec x cosec x dx dx = I= In n (cose(cosecx - cx - cotx) + ccotx) + c OR OR  In In tan ^ tan ^ + + cc OR OR -- In In (cose(cosecx + cx + cotx)cotx) xv) J

xv) J sinh sinh x dx = cosh x + cx dx = cosh x + c (xvi) (xvi) JJ cosh cosh xxdx dx = sin= sinh h x + cx + c (xvii) (xvii) JJ  sech  sech22x dx = tanh x + cx dx = tanh x + c xviii) J

xviii) J  cosech  cosech22x dx x dx = - = - coth x + ccoth x + c (xix) (xix) JJ sech sech xx . . tanh x dtanh x dx x = - = - sech sech x + x + cc xx)

xx) J cosechxJ cosechx. . cotcot hxdhxd x= x= -co-co sese chx + cchx + c   (xxi)(xxi) jj"" (ii)

(ii)

j

j

=

=--

/n/n (ax + (ax + b) b) + c+ c

ax+b a ax+b a r r 1 1 nn px+ px+ qq (iv) I (iv) I aPaPx+x+i i dx dx == — —  (a (a > > 0) 0) + + cc . . p p fnfn aa (vi) J

(vi) J  cos(ax  cos(ax ++bb)dx= - sin(ax)dx= - sin(ax+b+b )) + c+ c aa

(viii) f 

(viii) f   cot(ax+  cot(ax+b)b) dx dx = - = - /n /n sin(asin(ax+x+b)+b)+ c c aa

(x)

(x)  j cosec  j cosec22(ax(ax + b) + b) dx  dx = = _ _  1 1  cot(ax  cot(ax + b)+ + b)+ c c

dx dx . . , , x x ,, = sin = sin 11 — + c — + c xxii) xxii) J J „„dxdx „ „ = = -- tata nn-1-1 — — + c+ c aa22++ xx22 (xxiii) J(xxiii) J dx dx x-v/x x-v/x22-a-a22 aa 1 1 -l -l xx ^^ = = — —  secsec 11 —  —  +1+1 xxiv) xxiv) J J 7 7 =l=ln n ++ '' JJ _xx22 LL  +  + a a22 x

x22 + + aa22 OR OR    sinhsinh-1-1 - - + + cc aa xxv) J xxv) J xxvi) | xxvi) | dx dx Tn

Tn x x ++ OR OR    cosh"cosh"11 - - + + cc aa dx dx l l , , a+a+ xx In In 2 2 2 2 --a a -- x x 2a 2a aa -- xx + + cc (xxvii)(xxvii) jj dx dx 1 1 , , xx -- aa In In + + cc x x22-- aa22   2a 2a x+ax+a <!SBansal

(xxviii) j

.y

]a2-x2 dx = - A/a2-x2 + — sin"1 - + c 2 2 a (xxix) { *Jx2_+a2 dx= —  ^ x2+ a2 + — sinlr 1 - + c 2 2 a (xxx) j J x_v 2- a2  dx= - J x2- a2 - — cosh"1 - + c 2 v 2 a f e3*

(xxxi) I eax. sin bx dx = —-—x- (a sin bx - b cos bx) + c

a +b

f ax

(xxxii) J eax. cos bx dx = — — - (a cos bx + b sin bx) + c

a +b

3. TECHNIQUES OF INTEGRATION:

(i) Substitution or change ofindependent variable.

Integral 1= j f(x) dx is changed to }  f(4> (t)) f'(t) dt, by a suitable substitution x = (J) (t) provided the later integral is easier to integrate.

(ii) Integration by part: {  u.vdx=u j" vdx- {

(a) J v dx is simple & (b) /

0) d U _{f A} . vd? r1Y J f vdx r)v J du dx

function. Note: While using integration by parts, choose u & v such that du

dx

dx where u&varedifferentiable

dx is simple to integrate. This is generally obtained, by keeping the order of u & v as per the order of the letters in ILATE, where ; I-Inverse function, L-Logarithmic function,

A-Algebraic function,  T-Trigonometric function  & E-Exponential function

(iii) Partial fraction,  spiliting a bigger fraction_{into smaller} fraction _{by known methods.}

4. INTEGRALS OF THE TYPE:

 j  [f(x)]"f'(x)dx OR { dx putf (x ) = t & proceed

(") j J  _{ax' + bx + c} > 1 I ,_{yax + bx + c}dX > f VaxJ y 2 + bx + c <k

Express ax2 + bx + c in the form of perfect square & then apply the standard results.

("0 _Jf —?PX ^— dx, f i = q = dx .

ax2 + bx + c ' J  ^/ax2 + bx + c

Express px + q= A (differential co-efficient of denominator) + B.

(iv) j ex [f(x) + f'(x)] dx = ex. f(x) + c (v) J  [f(x) + xf'(x)] dx - x f(x) + c

f  dx

(vi) J _J n s N Take xn  common & put 1 + x n = t .

x(xn+l)

r

dx

(vii) J (n l) / n e N , take xn  common & put 1+x n = tn

2

(xn+l)n"^ x2'

(viii) dx c»(l₊ x f n

take xn common as x and put 1+ x n = t.

dx

OR

J

dx OR

J

dx

a+bsin z x " a+bcos 2 x asin 2  x+bsin xcos x + ccos 2 x

r r

Multiply N" & D by sec2x & put ta nx = t .

00 dx

a+bsin x OR

dx

a+bcos x OR

i

dx

a + bsin x + ccos x

(xii)

(xiii)

(xiv)

(xv)

Hint: Convert sines & cosines into their respective tangents of half the angles, put tan  — = t

a.cos x + b.sin x + c ^ Express Nr = A(Dr) + B — (Dr) + c & proceed .

 Lcos x + m.sin x + n dx

x2+ l

x 4 + K  x 2 +1 dx OR |

x2- l

 —; dx where K is any constant .

x + K x + 1 J

Hint: Divide Nr & Dr by x2  & proceed .

dx „ r dx

1

&

 J

₇

 _—t—;

_\— 

_i 

_;

_{put px + q = t .}

(ax + b) Vpx + q (ax2 + bx + cj ^px + q dx (ax + b) J- px + qx + r • j-u 1 ( dx • 1 , put ax + b = - ; . , , p u t x = -1 (ax + bx + cj yj px + qx + r  1 x - a  p - x

 — ^ °r  I V(x ~ a) (x ~ P) > put x = a sec20 - 0 tan20

dx or J - a) (p - x) ; put x = a cos20 + P sin20

dx

J(x - a) (x - p) ; put x - a = t

2

  or x - P = t2

(ilZ?a«sa/  Classes Definite & Indefinite Integration [18]

DEFINITE INTEGRAL

1 . f f(x) dx = F(b) - F(a) where J f(x) dx = F(x) + c

a

 b

VERY IMPORTANT NOTE : If F f(x) dx = 0 =>  then the equation  f(x) = 0  has atleast one

a

root lying in (a, b) provided f  is a  continuous function in (a, b) .

2 . PROPERTIES O F _{DEFI NITE INTEGRAL} :

 b b b a

P-L j  f(x) dx = J f(t) dt provided f  is same P - 2 _{f  f(x) dx = - J f(x) dx}

a a • a b

 b c b

P- 3 J f(x) dx= J f(x) dx+ J f(x) dx, where c may lie inside or  outside the interval [a, b] . This property

a a c

to be used when f  is piecewise continuous in (a, b).

a

P-4 J f(x) dx = 0 if  f(x) is an odd function i.e.  f(x) = -f(-x) .

= 2 j f(x) dx if  f(x) is an even function  i.e. f(x) = f(-x) . 0

 b b a a

P-5 J f(x) dx = j f(a + b - x) dx, In particular  J f(x) dx = J f(a - x)dx

P-6 j f(x) dx - J f(x) dx + J f(2a - x) dx = 2 J f(x) dx if   f(2a - x) = f(x)

0 0 0 0

= 0 if   f(2a - x) = - f(x)

na a

P-7 J f(x) dx = n Jf(x) dx ; where'a'is the period ofthe function i.e. f( a+x) = f(x)

0 0

 b+ nT b

P-8 _{j f(x) dx = J f(x) dx where f(x) is periodic with period T& n e l .}

a+nT a

na a

P-9 j  f(x) dx = (n - m) j f(x) dx if  f(x) is periodic with period 'a'.

ma 0

 b b

P-10 If  f(x) < ())(x) for a < X < b  then J f(x) dx < J 4> (x) dx

P - L L Jf (x)dx < ] I f(x) I dx.

P-12 If  f(x) >0 on the interval [a, b], then J f(x) dx > 0.

WALLI'S FORMULA:  ji/2

J sm-x. co s ^ dx - [Q -l )( n- 3) (n- 5 ) . . . . l ot 2 X ( m- 1 ) ( m- 3 ) . .. . l o r 2 j K

( m + n ) ( m + n - 2 ) ( m + n - 4 ) . . . . l o r 2

7C

Where K = if both m and n are even (m, n e N)

2

- 1 otherwise

4. DERIVATIVE OF AN TIDERIVATIV E  FUNCTION :

If h(x) & g(x) are differentiable functions of x then, m

- p 1 m d t - f [ h (x)]. h'(x) - f [g (x)]. g'(x)_{dx g(x)}

5. DEFINITE INTEGRAL AS LIMIT OF A SUM :

 b

J f(x) dx = Lmut h [f (a) + f (a + h) + f (a + 2h) + + f ( a + n-1T»)]

a

= h I f (a + rh) where b - a = nh

f=0

If a = 0 & b = 1  then, Limit h f (r h) = Jf (x) d x . w h e r e n h = l 0 R

Limit

v n ; r=i2 f f- - ] = J f(x) dx .vny o

6. ESTIMATION OF DEFINITE INTEGRAL:

 b

(i) For  a monotonic decreasing function in (a , b) ; f(b). (b - a) < J f(x) dx < f(a). (b - a) &

a  b

(ii) For a monotonic increasing function in (a, b) ; f(a).(b - a) < j f(x) dx < f(b).(b - a)

a

7. SOME IMPORTANT EXPANSIONS :

. . . , 1 1 1 1 , „ , .... 1 1 1 1 7T

0) 00 = In 2 (u) - 5 - + — T+~T+ — T+ 00 :

2 3 4 5 l2 22 3 4 6

1 1 1 1 7 t2 , . , 1 1 1 1 7 t2

(iii) — - + — - + 00=7— (iv) - r - + — + — + . — +

co=-1 2 3 4 12 1 3 5 7

, v 1 1 1 1 712

(v) —2~I 2-"1

22 42 62 82 24

 EXERCISE-I Q l l Q.4 J Q.6 j Q.9 j tan 20 Vcos6 9 + sin6 0 dx d0 _{Q 2 f} 5x 4  + 4x5 (x5+ x + l)2 K - ' f Q.5 Integrate j dx dx cos X 1 + tanxdx X ^ X2  + 2 X - 1

 by the substitution z=x + ^/x'2 +2x-1

I s

eJ +l x ^nxdx

a2 s in2x+b2 cos2x

4 - 2 i 4 2

a sin x+b cos x

Q.7 | cos 20 . Incos 9+sin 9

cos 0-sin 9 d0 Q-8 f - y ^J sin x +

dx sin ^x + sin 2x dx Q.IO J dx Q.12 f si n( x- a)d x V sin(x+a) (x + Vx(l + x) )2 Q.13 J (sinx)"11/3 (cosx)"1/3dx Q.14 j Q .l l J V x ^ + 2 dx cot xdx (1-sin x)(sec x+1) Q.15 j Q.17 | 1-Vx dx !l+Vx ^/x2+l[ln(x2+l)-21nx] Q.16 j sin"-l l + X dx X dx Q.18 j ln(lnx)+ (lnx) dx Q.19 j x+1

Q.20 Integrate ^ f'(x) w.r.t. x4  , where f(x) = tan-1x + /n^/l+x - / n ^ l ^

Q.21 j (Vx + l)dx Vx(Vx+1) Q.22 f -dx sin^ ,/cos3 ^

J:

4+xe*)2 dx dx Q.27 j ^Q-30

\

cosec x-cotx secx

 ]j _{cosec x+cotx^/l+2secx} dx cosx-sinx 7-9sin2x dx Q-23 J 3V(x + l )2(x-l) „ _ , r dx Q.26 j dx sinx+secx \2 x cosx-si nx dx

V

Q.33 f ^ S m X dx V sinx + V cosx

r ecosx(x sin3 x + cosx)

Q.36 j • v '

Q.3'9 j

Q.28 J tan x.tan 2x.tan 3x dx

r 3+4sinx+2cosx Q.31 J :   3+2sinx+cosx Q.34 f — ^J _{sin x +1} secx+cos ecx dx dx Q.29 j —  f  — sinx Jsin(2x + a) . ^n(cosx+Vcos2x] Q.32 [ —^ _J ~ dx sm x -tan x 2 Q 35 \ 3x 2 +1 ( x2- l )3 dx sin2 x dx r (ax -b) dx Q '3 7  J xVc2x2-(ax2+b)2 r e \z—a Q.38 /= (l-x)Vl dx

 ,„

2_\3/2 _dx (7x-10-x ) Q.40

J

xlnx dx Q.42 j 2 - 3 x j1+x 2 + 3x (^J } E i x q.43 f f " " dx Q.44 { 1- x J 1 + 3sm2x J Q.41 f J i z * dx v J   Ali+x X 4x5 - 7x4 + 8x3 - 2x2 + 4x - 7 x2(x2 +1)2 dx

Q4 5 1 ( 2_{(x +3x+3 j^/x+1}XI2 V> r ~ T Q.46 j ^ -J x - x 2 dx Q.47 J x2 J  (x-a)V(x-a)(x-(3) ^ r dx f  Jcos2x <• (l + x2)dx Q.48 Y Q.49 f — dx Q.50 I S — r a e ( 0 , TC) x3V(l + x)3 v J si n x v J  l - 2 x2c o s a + x4 V J

 EXERCISE-II

71

q'!^™ 

Q2

_IV 

_T

_H3I

dx

QJ

_!

 /n3xdx

0

 ji/2 TI/2 TI/4

x dx

Q.4 J sin2x. arctan(sinx)dx Q.5 J cos43x. sin26xdx Q.6 j _

o o • o  c o s x  (cosx + sinx)

Q.7 Let h (x) = (fog) (x) + K where K is any constant. If (h(x)) = ^ ^ — then compute the

dx cos (cos x)

f(x)

value of j (0) where j (x) = j" —— dt; where f and g are trigonometric functions.

g ( x ) S W n/2

Q.8 j 7(co s2n_1x - cos2n+1x) dx where n e N

- i t / 2

Q.9 Evaluate the integral: J (jx  + 2 yflx  - 4 + Jx-2 ^2x^4  j dx

3

00 00 oo

i x2 r xdx r dx

Q.10 I f P = - d x ; Q = and R = T then prove that

 j  l + x J  1+ X j  1+ X 0 0 0 (a) Q= (b) P = R, (C) P - V 5 Q + R = ^ Q.ll Provethat ? . ( n - l K a , h)x +n a h ) ^ ^ bn . _ an-! J a (x+ a) (x-t- b) 2(a + b) l   x(l-x)r x i i - x 4 , f vf x22l n v.In x , r yf  x 2" x Q.12 J , dx q.13 J 2LJE2L dx Q. 14 Evaluate: J 7 2

0

1+ X

0

 Jl-x2

_2"VX +4

^ n/2 271 Q.15  f ^ - f j d x   Q.1 6 J d x Q. 17 j 0 1 + x 0 sm(f +xj ^ J0 dx 2-n , dx 2+sin2x 2ti / \ ' 7T XN Q .1 8 J e x  cosf  — + — dx Q 1 9 f  2X7+3X6-10X5-7X3-1 2X2+X+1 V2 X2+ 2 dx v4 2/

Q.20 If for  non-zero x, a f(x) + bf —j = f — I - 5; where a ^ b then evaluate j  f(x) dx. vx

 ji/4

Q.21 J  c o s x-s m xd x Q.22 f   ( a X + b ) s £ C X t a n X  dx (a,b>0)

0 10 +sin 2x v J0 4 + tan x V' 7

Q.23 Evaluate: f ( 2 x + 3 ) f x d x . I (1 + cos x)

3/2

 p+ qn

Q.24 J |x. sinTtxl dx

Q.25 Show that J | cos x| dx = 2q + sinp where q s N  & -— < p <—

o 2 2

"5 2/3

Q.26 Show that the sum ofthe two integrals J e(x+5^dx+3 J Q9(x-2/3f dx is zerQ

it/4 Q.27

I

smx+cosx "o c os2x+sin4x 1 + x x3 1/3 71/2 dx Q.28 J _{* q <am V4.h rr\c y}r 2— 72sin x — dx (a > 0, b > 0)v ' ' 0 n/2  _ f . I -t- X X. Q.30

j

a sin x+b cos x Vl+sinx+ .^1-sinx tan" ^/l+sm x - ,/l-sin x dx x . d x  p « .1.UA Q.31 (a) J Vto^dx ; (b) J // 2 2v,2 H ^ Vlx -a Ab ) 2 1

Q.3 2 j | x - t ! .cos7it dt where 'x' is any real number o Q.34 Provethat J = J - J * (n> 1) 0 1 + x 0 ( l- xn) X X Q.35 Showthat Je"-e"z i dz = ex^4f e ' ^4 dz o o Q . 3 7 ( a ) | l ^ , J» - , (b) f  d x 2a Q.33 j x sin-1 1 2 a - x 2 V a dx Q.36

J

r x

2 _{sin 2x,sin (y.cos x )}

2 x - 7 t dx o l+x'Vx+x^x3 ' 0 1 + x 2 ' ^ OO 1 1 Q.38 Showthat f — — =2 o x + 2 x c o s 9 + 1 n d x 0 x2  + 2xcos9 + 1 sinB if 0 s (0,ic) 9 - 2 7i . sin 9 if 0 g ? dx 271 (a+b)

Q.39 Prove t h a t j (x2+ax+a2) (x2+bx+b V V^ab (a2+ab+b2) '

Q.40

j

x 2 (sin2x-cos2x) (1+sin 2x)cos2 x dx X , s, X r  u Q.41 Prove that J f f (t) dt n J VO J Q.42 J d x o (5 + 4 cos x)2 16 du= J f(u).(x-u)du. o Q.43 Evaluate J /n(Vl-x + Vl + x)dx 16 27t . 2 Q.44 (a) j t a n " 1 d x , (b) Evaluate J d0 a > b > 0

(ilZ?a«sa/   Classes  Definite & Indefinite Integration  _[18]

 A

Q.45 Let  f(x) = - j /n cos y dy then prove that  f(x) = 2f

v4 2, - 2 f 71 X 4 _ 2 -x/n2. J t / 2 Hence evaluate J s e c t dt ~ . ^ , r x. lnx , , r a x. dx Q.46 Showthat J f ( - + - ) . dx = ln a. Jf (- +- ). x a x x a x

Q. 47 Evaluate the definite integral, j

1_-(2x332 _+x99 8 _+4x1668_-sinx69 1₎ -l 1 + x666 dx Q.48 Prove that ( a ) } V ( x - a ) ( P - x )

0»

IM

 p (c)5 d x { xA/ ( x - a ) ( ) 3 - x ) V ^ P  p _x.dx Q.49 If f(x) = 4 cos x 1 1

(cosx-1)2 (cosx + 1)2  (cosx-1)2

(cosx + 1)2 (cosx + 1)2 cos2x

1

/ n t a n- 1_{x - 1 ,}

, find Jf( x) dx

Q.50 Evaluate:

Je

/ntan_ x-sin _1(cosx)dx.

o

 EXERCISE-III

Q.l If the derivative of f(x) wrtx is — - then show that f(x)is a periodic function.

f ( x )

Q.2 Find the range of the function,  f(x)= f  Sm X —7 .

1 —   2t cosx + t

Q.3 A function  f is defined in [-1,1] as f (x) =  2 x sin —   - cos — ; x ^ 0 ; f(0) = 0 ;

X X

f (I/71) — 0 .  Discuss the continuity and derivability of f at x=0. - 1 if - 2 < x < 0

Q.4 Let f(x) = [ I and g(x) = J f(t) dt. Define g (x) as a function of x and test the

x -* 1 if 0 < x < 2 _2

continuity and differentiability of g(x) in (-2,2).

Q.5 Prove the inequalities :

(a) 0 < J 0 2s x7dx_ 1 /3 8

M l

(b) 2 e~ 1/4 < } ex2"x dx < 2e2.

(c) a< f — < b then find a & b. (d) ^ < J

i 10+3cosx — 2 0

dx

2 + x2 " 6

Q.6 Determine a positive integer n<5, such that J ex(x- l)n dx = 16 - 6e.

Q.7  Using calculus

(a) If  | x | < 1 then find the sum of the series _ L + 2 x + 4 x 3 + 8 x ? + QO .

1 + X 1 + X2 1 + x4 1+ x8

/UN 1,1 I / ! .U * l - 2 x 2 x - 4x3 4x3- 8 x7 1 + 2X

(b) If |x| <1 provethat - + r  r  + r + .00 = - .

1 - x + x 1 — x + x 1 - x + x l +x + x2

x 1 x l x l x

(c) Prove the identity f (x) = tanx + - tan + t a n^ j + + 77 tan = — 7 cot ^77 - 2cot 2x

Q.8 If   (j)(x) = cos x - J (x -1)  cj)(t) dt. Thenfindthe value of   <|>" (x) + cj)(x).

2 a 0 " dx X  jlntdt Q.10 If y = x1 , find 7^- at x = e. dx 1 x d2

Q.9 If  y = - f f  (t) • sin a(x - 1) dt then prove that —^ + a2y =/(x).

9 * A^t*

1

Q. 11 If  f(x) = x + j [xy2+x2y] f(y) dy where x and y are independent variable. Find f(x).

dx

Q. 12 If f (9) = —_w

de  _{I -~ COS0 co sx} . Show that f ' (9) . sin 9 + 2 f(9). cos 9 = ~

r f' fx")

Q. 13(a) Let g(x) = xc. e2x & let f(x) = J e2t. (312 + 1)1/2 dt . For a certain value of'c', the limit of — ^

0 g' (x)

as x ->• 00 isfinite_{and non zero. Determine the value of'c' and the limit.}

f t2 dt J JaTt

(b) Find the constants 'a' (a > 0) and 'b' such that, Lim  — = 1.:

x-»o b x - si nx

Q. 14 If f: R - > R is a continuous and differentiable function such that,

x O x 2 x

J

f (t) dt + f ' " (3) | dt -

j

(t3) dt - f '

(1) J

(t2) dt + f " (2)

J

(t) dt

-1 x 1 X 3

thenfind_{the value of f ' (4).}

Q. 15 Given that Un= {x(l - x)}n & n > 2 prove that - n (n - 1) Un _ 2- 2 n(2n - l)Un _  p

1

further  if Vn - J ex. Un dx, prove that when n > 2, Vn+ 2n(2n- l). Vn _ r  n(n- 1) Vn _ 2 = 0

0

J?nt  %£n2

.2

equation.

" 1 - x if 0 < x < l

Q.17 Let f(x)= 0 if l < x < 2 • Define the function F(x) = J f(t) dt and show that F is

(2-x)2 if 2 < x< 3 °

continuous in [0, 3] and differentiable in (0,3).

f Ait 71 2

Q. 16 If  J —2—2 = (x > 0) then show that there can be two integral values of 'x' satisfying this

X ~F~ T 4

0

Q.18 Let f  be an injective function such that f(x) f(y) +2 = f(x) + f(y) + ffay) for all non negative real x& y w ±th £' (0) - 0 & f'' (1) = 2 # f(0). Find f(x) & show that, 3 J f(x) dx - x (f(x) + 2) is a constant.

Q.19 Evaluate : (a) ^  A

1/n (b)  L i m 1+- 1+- _{1 + -} 1+ -l/n (c) Lim I _{n+1 n+2} 1 + 2 + + —_4n3n . . . . . b f  dx 1 1

(d) Using ab initio prove that J — = —  - —

Q.20 Prove that sinx + sin 3x + sin 5x + .... + sin ( 2k- 1) x: sin

 2 kx k e N and hence sinx 71/2  prove that, j £ H L ^d x =: i + - + _ + ^ + 1 1 1 + 1 sin x 3 5 7 . 2 k-l "J? • 2

Q.21 If Un= dx, then show that U,, U2, U3, , Un  constitute an AP .

q sin x

Hence or otherwise find the value of Un.

Q. 22 Solve the equation for y as a function of x, satisfying

 X X

x • J y (t) dt = (x +1) j t •• y(t) dt, where x > 0.

o o

1

Q.23 Prove that: (a) Iffl n = j xm. (1 - x)n dx =

m!n! ( b> Im ; I 1= j xm. ( l n x )ndx = (-l )n (m+n+1)! n! m, n s N. m, n s N. ^n+i o (m+1

Q. 24 Find a positive real valued continuously differentiable functions/on the real line such that for all x

 f

 X

!

( x ) = / ( ( / ( t t f + C / ' W ^ j d t + e2

Q.25 Let f(x) be a continuously differentiable function then prove that, J [t] f' (t) dt = [x], f(x) - V f (k)

i k=i

where [• ] denotes the greatest integer  function and x > 1.

Q.26 Let f be a function such that  | f(u) - f(v) I < | u - v | for all real u & v in an interval [a, b]. Then:

(i) Prove that f is continuous at each point of  [a, b].

(ii) Assume that f is integrable on [a, b]. Prove that,

7t f dx

Q.27 Establish the inequality : — < I 7 = = f

6 o V 4 - x - x

f f(x) dx - (b -a ) f(c) (b-a)

2

, where a < c < b

<

Q. 28 Show that for a continuously thrice differentiable function f(x) f (x) -f (0 ) = x f ( 0 ) + ® ^ - + i }f "' (t )( x- t)2dt

2 o

n / v I m 1

Q.29 Prove that £ ( " l )_{k = o} k  k 1 — T = S ( " ^ ( i f ) ,VK/ _{k + m + 1}

k = o \ k / k + n + 1

Q.30 Let / and g  be function that are differentiable for all real numbers x and that have the following  properties: (1) /' (x )= /( x) -g (x ) (ii) g'(x)=g(x)-/(x) (iii)  f ( 0 ) = 5 (iv) g(   0 ) = 1

(a) Prove that/(x) + g  (x) = 6 for all x.

(b) Find/(x) and g  (x).

 EXERCISE-IV

Q 1   Ftad S J s _n , i f : S _n = ^ _{+ T} ^ _{+ 7} 4 = ₊ +  — = = = = .  [REE'97,6] V4n -1 y4n -4 V3 n'+2n-l X

Q.2 (a) If g (x) = Jc o s 4 t d t , then g (x + %) equals :

(A)g(x) + g(7C) (B) g(x)-g(7C) (C)g(x)g(7C> (D) [g(x)/g(7l)] (b) Limit i g _ ^ e q u a l s : r=l Vn2+r 2 (A )l + V5 (B) - 1 + VJ ( Q - 1 + V2 (D )l + V2 e" (c) The value of J ————— dx is I H esinx 4 ?

(d) Let — F(x) = , x > 0 . If f — dx = F (k) - F (1) then one ofthe possible

dx x ' x

values ofki s .

(e) Determine the value of J 2 x 0 + s i n x) d x [JEE '9 7, 2 + 2 + 2 + 2 + 5]

„ 1 + COS X

Q.3 (a) If ff(t) dt = x + f t f ( t ) dt , then the value of /( l) is

(A)° 1/ 2 (B) 0 (C) 1 (D) - 1 / 2

i f  Y  \ i

(b) Prove that f tan"1 dx = 2 f tan 1x dx . Hence or otherwise, evaluate the integral

i U-x+x2; i

i

Jtan _1(l-x+x2)dx [JEE'98,2 + 8]

Q 4 E r a l u a t e [ R E E ' 9 8 - 6 1

Q.5 (a) If for al real number  y, [y] is the greatest integer less than or equal to y, then the value of the 371/2

integral j [2 sinx] dx is:

(b) (A) - n (B) 0 3,1/4 _rl Y f — — — is equal to : L 1+ cosx (A) 2 (B) - 2 (C) 2

(C)

(D) * 2 / N T R  x3+ 3x + 2 , (c) Integrate: J — dx (x2+l ) (x + 1) ii

(d) Integrate: J _{„cosx . -cos x}

e +e -dx

71/6

Q.6\\ Evaluate the integral J V3 c o s 2 x

cosx

[JEE '99, 2 + 2 + 7 + 3 (out of200)]

[REE'99, 6]

Q.7 (a) The value of the integral j logex dx is:

(A) 3/2 (B) 5/2 (C) 3 (D) 5

 x 

r 1 ' 1

(b) Let g( x) = J f(t ) dt, where f is such that < f( t) < 1 for t £ (0, 1] and 0< f(t) <

-o 2 2

for t G  (1,2]. Then g(2)  satisfies the inequality:

( A ) - | < g ( 2 ) < I (B) 0 < g (2) < 2 ( C ) ^ < g ( 2 ) < | ( D ) 2 <g ( 2) < 4

(c) If f (x) = {

(A) 0

ecosx . sin x for |x| < 2

2 otherwise , (B) 1  j Then j f(x)dx : (C) 2 (D)3 x t> t

(d) For x > 0, let f (x) = j dt. Find the function f(x) + f (1/x) and show that,

f(e) + f( l/ e) = 1/2. Q.8 (a) Sn - — l~r= + 1 1+t + + [JEE 2000, 1 + 1 + 1 +5] n + Vn Find   L i m i t _{S_} f sin t r u

(b) Given d t = a , find the value of —  d t in terms of a .

0 1 + t 471 - 2 4 7C + 2 t

[ REE 2000, Mains, 3 + 3 out of 100]

471 sm n—> co n t 2x + 2 Q.9 Evaluate f sin 1 W 4 X2+ 8 X + 13 IT/: Q.10 (a) Evaluate } dx. it/2 9 51 cos x r xdx  — 5 r r ~d x. (b) Evaluate J :

0 cos3 x + sin3 x / 0 1 + cos a sinx

[REE2001, 3+ 5]

X

Q. 11 (a) Let  f(x) = j  ^ 2 - t2 dt Then the real roots ofthe equation x2- f ' (x) = 0are

(A)±l (B) ± - "" ( C ) ± - (D) 0 and 1

(h) Let T > 0 be afixedreal number. Suppose/ is a continuous function such that for all x e R

T  3+3T

/ ( x + T) = /( x) . If I = f /f x; dx then the value of J ' j W dx is

0 3

(B)21 (C) 3 I (D) 61

(A) | I

(c) The integral JI [x] + / n j j - ^ J dx equals

( A ) - - (B)0 (C)l (D) 2/n (d) For  any natural number m, evaluate J (x3m + x2m + xm) (2x2m +3xm + 6)m dx, wher ex> 0 [JEE 2002(Scr.), 3+3+3] [JEE 2002 (Mains),4] n/2 n/4

Q.12 If f is an even function then prove that J f (cos2x) cosx dx= V2 J f (sin2x) cosx dx

0 0

[JEE 2003,(Mains) 2 out of 60] l

Q-i3

(

a

)

f ^

d x = o ( A ) f  + 1 ( B ) f - . (C)7t (D)l 0 (b) If jx f( x) dx =- ^t5, t > 0 , t h e n f 5 r 4 \ v25y 2 ( A )7

(B)

( O - f 2

(c) If y(x) = f   c o s x c°s ® DQ ^en find at x = TC.

2J, i+sin2 Ve d x (d) Evaluate J -dx. -it/3 2- co s X + -TC Q.14 (a) If Jt2(f(t))dt =(1-sinx),then/ sinx (A) 1/3 (B) i/V3 vV3y is (C)3 (D)l [JEE 2004, (Scr.)] [JEE 2004 (Mains), 2] [JEE 2004 (Mains), 4] [JEE 2005 (Scr.)] (D)V3

(b) J (x3+ 3x2 + 3x + 3 + (x +1) cos(x + l))dx is equal to

- 2

(A )- 4 (B)0 (C)4

 f

(c) Evaluate: fe|cos^ 2sin

0 V -cosx v2 ,+3 cos x' N  —  cosx v2 j j sin x dx. [JEE 2005 (Scr.)] (D)6 [JEE 2005, Mains,2]

Q15

  ^W5?^T

dXisequalt0

( A ) V 2 x4 - 2 x 2+ l + c

(B)

(C) x - 2 x z + l + C V2X 4 -2X2+1 2x2 + C [JEE 2006,3] Comprehension

^ ^

Q.16 Suppose we define the definite integral using the following formula (f(x)dx = (f (a )+ f( b) ), fo r

more accurate result for c e (a, b) F(c) = ^ (f (a) + f (c)) + — - (f (b) + f (c)). When c = p p ,

2 2 ^  b i_  b - a Jf (x)dx = ^ ( f ( a ) + f ( b ) + 2f (c>) a i t / 2

(a) | sin xdx is equal to

o

(A

) | ( l + V2) + ( C ) ^71

_(D)

71

4V2  jf ( x ) d x _ i _ i (f ( t ) + f (a))

(b) If/(x ) is a polynomial and if Lim—

t->a 0 for all

a

 then the degree of 

 f 

 (x) can

atmost be

(A)l  (B)2  (C)3  (D)4

(c) If /"(x) < 0, V x s (a, b) and c is a point such that a < c < b, and (c,/ (c) ) is the point lying on the curve for which F(c) is maximum, then/'(c) is equal to

(A) f(b)-f(a)  b - a 2(f(b)-f(a))  b - a (C) 2f(b)~f(a) 2 b - a 5050} ( I - x5 ° R dx

Q. 17 Find the value of

J ( I- x5° R <

(D)0

[JEE 2006, 5 marks each]

[JEE 2006,6] dx

 ANSWER KEY

 EXERCISE-I

Q.l In l + Vl + 3cos 2 20 cos 20 + C ^ _ x + 1 Q.2 -1 x j o x 3

Q.3 - /n(cosx + sinx) + - + - (sin 2x + cos 2x) + c Q.4 - tan-1 x - ——  7 - — In

2 o 0 Q.5 2 tan-1  (x+Vx2+2x-l) + c  —  I all X - / . \ - — 8 4(x - 1 ) 16  f _{x - 1}iN VX + V + c Q.6 I* .e/ - - + c

Q.7 (c) - (sin 20 ) In cos9 + sinG ] 1 cos9 - sin9j 2 1 - - /n( sec2 0) + c Q.8  - In tanx tanx+2 + c f / x + tan V Q.ll J (x+V^")'

Q.12 cos a. arc cos

-1 a tanx  JJ + c Q.IO + w hent = x + V ^ (x+VX2+ 2 1/2 f _cosx\ vcosay

- sin a 1 ( • In sinx + Jsin ^x-sin 2 2""^ , +C ^ Q.13 —^ 3(l+4tan x)tt t1

Q.14  - In tan- + - sec, 1 2, x - +ta n - + cx

Q.16 (a + x) arc tan - ^ax + c

Q.18 xln (lnx) - ——  + c lnx 8(tan x) Q.15  V^Jl^x-2-Jl^ + arc cos Vx+c (x2+l \l + c Q.17 Q.19 In /x2 +1 9x xe" vl+xex j 2-31nfl+-^ l+xex V x + c Q . 2 0- / n ( l - x4) + c Q.21 6 t 4 t2 1  — - —  +1 + ~/n( l + t  z  ) - ta n-11 4 2 2

Q. 2 2 . +2tan"1 Jcosf   -In 1 + -yjcOSj

ycosf  v 1  - ycosi + C where t = x1/6 + c Q.23 -v x - l y + C  Q24. sin" 1  —sec — v.2 2 j ~ 1 , (4+3sin x+3cos x) Q.25 —In- —+c 24 (4-3sin x-3cos x) Q 27 1 VT+sin x-cos x

Q . 2 « 2 sin x - cos x —p=ln tan

V2

F

X 71 ^

2V3 V3 -sm x+cos x+ arc tan (sin x+cos x)+c  Q.28 -^n(secx) - ^ ^n(sec2x) +^n(sec3x) + c

Q. 29- 1  In

V sina Q.31  2x-3arctan

cotx+cota+^/cot2x+2cotacotx-l + c  Q.30 x sin x + co sx— + c x c o s x - s m x

' x ^

tan—+1

v 2 ,

+c _Q.32 Vcos2x

smx - x - cotx . In (e (cosx + Vcos2x)j + c

(ilZ?a«sa/   Classes  Definite & Indefinite Integration

_[1

_8]

7

1 „ 1 t2 -  -Jit  + 1 1 . - , —

Q.33 /n(l +1) - - /n(l +1*) + ^ /n t2 + + 1 - j t a n *  + c w h e r e 1 = Vc o t x

 _ 1, X 1 2X

Q.34 —lntan tan —+c

2 2 4 2 Q.35 c - ( x2- l )2

Q.36 c - ecos x(x + cosec x) Q.37 sin-1 ax +b

cx

J +k Q.38 e x J ^ + c Q.39  20>> +c  _ lnx Q.40 arcsecx—, +c

i

I 2— Q.41 In , ' + V3tan Vu4+u2 + l /r l+2u

V3tr~

1

9^7x-10-x

2 V3 + c  where u = 3V

 1+x

Q.42 -

tan t + —

1 . f  V51 - 1r=

 

~7=

2V5

vv5t + 1

- |sin 1 x  - ^/l-x2 j  + c where t = h^-

1+x

x

Q.43 tan

.^2 sin 2x

v

sinx + cosx

y + c Q.44 4/nx+-+6tan-7 1( x) + 6xj +C X I T X 2 x  J 2 - X - X2 V 2 „ f 4 x + 2 V 2 J 2 x -Q.45 —i=arctan—p —+c Q.46 + — V3 V^+i) x 4

x

x

.

(2x+l

sin

^ 3

+ c - 2 x ^ Q . 4 7 — - J — - + c a - p y x- a 1 4 - - 2 15 , Q.48 n x x 2 + — 

In

4x

2

y l + x

8 V T + x - i 7 1 + X + 1 + c Q.49 V2 nl V2-t - —/n2 ' 1 - 0

vl + ty where t = cos9 and 9 = cosec' (cotx)

Q.50 cosec—a 2 tan

-1

\ v , 2 x y a cosec— 2_/

 EXERCISE-II

2 ^ Q.l — Q.2 /n2 Q.3 6 - 2e Q.4 - - 1 Q.5 Q.6 £ In2 ft 2 64 8 Q.7 1 - sec(l) Q.8 — Q . 9 2 V2 + | (3 V3 - 2V2) Q.12 2n + l f 22 ^ 7t 17 J 7t Q . 1 3 - ( l - l n 4 ) Q.14 4V2-4/n(V2 + l) Q.15 ^ Q.16 Q . 1 7 ^ = Q . 1 8 - ^ ( e 2 * + l )  _ 71 16V2 _ (aln2-5a+^) Q.22 (a7t

+ 2b

)7t 3V3 Q.23 7t(7t + 3) Q.24 Q.21 i

3

371+1

7t2 V2 l"

arc tan arc ta

n-3

3

Q 27

Q.28 71 2a(a+b) 57t 3 Q.29 Q.30 ^ L Q.31 (a) ^=[rc+21n(V2-l)] (b) 16

12

2

2

2

2

Q.32—5- cos7ixf orO<x < 1; for x> 1 & — 2 f o r x ^ ° Q -3 3 —

71 71 71 4 Q . 3 6 -% Q. 37( a) f ( b ) ^ Q.40 ^ - 7 /n2 Q.42 ^ 16 4 27 Q.43 - In 2 +2 1 167t _  rr   271 Q.44 (a) — 2V3 (b) i - J ^ b2 Q.45 - / n 2Tt Q.47 71+ 4 666 32 Q.49 -2ti - — Q.50  — ( l + / n 2 ) + -8 4 2  EXERCISE-III , 71 TC ,

Q.2 - - , - (• Q.3 cont. & der. at x = 0

Q.4 g(x) is cont. in ( - 2, 2); g(x) is der. at x = 1  & not der. at x = 0 . Note that; -( x + 2) for - 2 < x < 0 g(x) = 2 2TI 27T -2 + x - ^ - for 0 <x< 1 Q.5 a= — & b = — Q. 6n = 3 ^ - - x - 1 f o r   1 <x< 2 Q.7 (a) -—- Q.8 - cos x Q . l O l + e Q. ll f(x) = x + 1—x 119 119  R

Q.13 (a) c = 1 and Limit  w i l l b e ( b)a = 4a nd b =1 Q.14 10

if 0<x<l 2 x2 X - T Q.16 x = 2 or 4 Q.17 F(x)= \ if l<x<2 k if 2<x<3 M . 3 1 2 Q.18 f(x) = l + x2 Q.19 (a) - (b) 2e(1/ 2)("-4) (c) 3 - / n 4 Q.21 Un= ^ e 2 Q.22 y = ^ e"1/x Q.24 f  (X) = ex +1  Q.30 f (x) = 3 + 2e2x; g (x) = 3 - 2e2x