definite and indefinite integration

7/23/2019 Definite and Indefinite Integration

http://slidepdf.com/reader/full/definite-and-indefinite-integration 1/18

K E Y C O N C E P T S

1. D E F I N I T I O N :

If f & g are functions of x such that g'(x) = f(x) then the function g is called a PRIMITIVE O R

ANTIDERIVATIVE O R INTEGRAL of f(x) w.r.t. x and is written symbolically as

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

dx

2. STANDARD RESULTS :

\ n + lr (ax+b)n

i) (ax + b)°dx= , \ + c n * - lJ a(n+l)

iii) f eax+b dx = - eax+b + ca

v) f sin(ax+b) dx = -— cos(ax +b) + ca

vii) j tan(ax + b) dx = — In sec (ax + b) + ca

ix) J sec2 (ax + b) dx = — tan(ax + b) + ca

xi) J sec (ax + b) . tan (ax + b) dx = — sec (ax + b) + ca

xii) J cosec (ax + b) . cot (ax + b) dx = cosec (ax + b) + ca

xiii) J secx dx=In (secx + tanx) + c OR In tan { + y j + c

xiv) J cosec x dx = In (cosecx - cotx) + c OR In tan ^ + c OR - In (cosecx + cotx)

xv) J sinh x dx = cosh x + c (xvi) J cosh xdx = sinh x + c (xvii) J sech2x dx = tanh x + c

xviii) J cosech2x dx = - coth x + c (xix) J sech x . tanh x dx = - sech x + c

xx) J cosechx. cot hxd x= -co se chx + c (xxi) j"

(ii) j =- /n(ax + b) + cax+b a

r 1 n px+q

(iv) I aPx+i dx = — (a > 0) + c. p fn a

(vi) J cos(ax+b)dx= - sin(ax+b) + ca

(viii) f cot(ax+b) dx = - /n sin(ax+b)+ ca

(x) j cosec2(ax + b) dx = _ 1 cot(ax + b)+ c

dx . , x ,= sin 1 — + c

xxii) J „dx

„ = - tan-1 — + ca2+x 2

(xxiii) Jdx

x-v/x2-a2 a

1 -l x ^= — sec 1 — +1

xxiv) J 7 =ln +' J Lx2 + a2

x 2 + a 2OR sinh-1 - + c

a

xxv) J

xxvi) |

dxTn x + OR cosh"1 - + c

a

dx l , a+xIn2 2 -a - x 2a a - x

+ c (xxvii) jdx 1 , x - a

In + cx 2 - a 2 2a x+a

<!SBansal Classes Definite & Indefinite Integration [2]



(xxviii) j .y]a2-x2 dx = - A / a2 - x

2 + — sin"1 - + c2 2 a

(xxix) { *Jx2+a2 dx= — ^x2 +a 2 + — sinlr 1 - + c

2 2 a

(xxx) j J x 2 - a 2 dx= - J x 2 - a 2 - — cosh"1 - + c

v 2 v 2 a

f e3*

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

f ax

(xxxii) J eax. cos bx dx = — — - (a cos bx + b sin bx) + ca +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 vdxr)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 > 1 I , dX > f Vax2 + bx + c <kJ ax' + bx + c yax + bx + c J y

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

("0 f —?PX

^— dx, f i = q

= dx .J

ax

2

+ bx + c '

J

/ax

2

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

(iv) j ex [f(x) + f'(x)] dx = e

x. f(x) + c (v) J [f(x) + xf'(x)] dx - x f(x) + c

f dx(vi) J n s N Take xn common & put 1 + x

n = t .J x(xn+l)

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

2(x

n+l)

n"^x

2'

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



(viii)dx

c»(l + x f n

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

dxOR J

dxOR 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 tanx = t .

00 dxa+bsin x

OR dxa+bcos x

OR i dxa + bsin x + ccos x

xi)

(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 +1dx OR |

x2

- l —; dx where K is any constant .x + K x + 1 J

Hint: Divide Nr & Dr by x2 & proceed .

dx „ r dx1 & 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 sec2 0 - 0 tan2 0

dx or J - a) (p - x) ; put x = a cos2 0 + P sin2 0

dx

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

2 or x - P = t

2




DEFINITE INTEGRAL

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

b

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

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

(m+n)(m+n-2) (m+n-4) . . . . lo r2

7CWhere 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 = nhf=0

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

Limitv n ; r=i

2 f f- - ] = J f(x) dx .v

ny 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 7T0)

00 = In 2 (u) - 5 - + — T + ~ T

+ — T

+ 00 :

2 3 4 5 l2 2

2 3 4 6

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

2

(iii) — - + — - + 00=7— (iv) - r - + — + — + . — + co=-1 2 3 4 12 1 3 5 7

, v 1 1 1 1 71 2

(v) —2~I 2-"

1

22 42 6 2 8 2 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 f5x4 + 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 cos2x4 - 2 i 4 2

a sin x+b cos x

Q.7 | cos 20 . Incos 9+sin 9

cos 0-sin 9d0 Q-8 f - y ^

J sin x +

dx

sin ^x + sin 2x

dx Q.IO Jdx

Q.12 f sin(x-a) dx

V sin(x+a)

(x + Vx(l + x) )2

Q.13 J (sinx)"11/3

(cosx)"1/3

dx 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(x

2+l)-21nx]

Q.16 j sin"-l

l + X

dx

X

dx Q.18 j ln(lnx)+(lnx)

dx Q.19 jx+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

7-9sin2xdx

Q-23 J 3V(x + l )2 (x-l)

„ _ , r dxQ.26 j dx

sinx+secx

\ 2

x cosx-si nxdx

VQ.33 f ^ S m X dxV 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+2cosxQ.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 [ —^ ~ dxJ sm x

-tan x

2

Q 35 \3x2 +1

(x2- l )

3 dx

sin2 xdx

r (ax -b) dxQ

'3 7

J xVc2x

2-(ax

2+b)

2

r e \z—aQ.38 /=

(l-x)Vldx

,„ 2\3/2 dx(7x-10-x )

Q.40 J xlnxdx

Q.42 j2 - 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 * dxv J Ali+x X

4x5 - 7x4 + 8x3 - 2x2 + 4x - 7

x2(x2 +1)2dx

^Bansal Classes Definite & Indefinite Integration m



Q 4 5

1 ( 2XI 2 V> r~T Q.46 j ^ - x - x 2 dx Q.47 J(x +3x+3 j^/x+1 J x2 J (x-a)V(x-a)(x-(3)

^ r dx f Jcos2x <• (l + x2)dxQ.48 Y Q.49 f — dx Q.50 I S — rae(0 , TC)

x3V(l + x)3 v J si n x v J l - 2 x 2 c o s a + x 4 V J

EXERCISE-II

71

q'!^™ Q2IV TH3I dx QJ ! /n3xdx0

ji/2 TI/2 TI/4x 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 thedx 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(cos2n_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 dx3

00 00 oo

i x2 r xdx r dxQ.10 IfP= -dx ; 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 - V5 Q + R = ^

Q.ll Provethat ? . ( n - l K a , h)x + n a h ) ^ ^ b n . _ an-!Ja (x+ a) (x-t- b) 2(a + b)

l x(l-x)

4 f x2 .In x , f x 2 " xrx i i - x , f v2 l n v r yQ.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 j0

1 + x 0 sm(f+xj ^

J0

dx

2-n ,

dx2+sin2x

2ti / \' 7T X

N

Q . 1 8 J ex cosf — + — dx Q 1 9 f 2X7+3X

6-1 0X5-7X

3-1 2X2+X+1

V2 X2

+ 2

dxv 4 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 x d x Q.22 f ( a X + b ) s £ C X t a n X dx (a,b>0)0 10 +sin 2x v J

0 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 2/3Q.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 Ismx+cosx

"o cos2x+sin4x

1 + x x 3

1/3

71/2

dxr sin x

Q.28 J 2— 72 — dx (a > 0, b > 0)* q <am V4.h rr\c y v

' '0

n/2 _ f . I -t- X X.

Q.30 j

a sin x+b cos x

Vl+sinx+ .^1-sinxtan"

/l+sm x - ,/l-sin xdx

x .dx p « .1.UAQ.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 numbero

Q.34 Provethat J = J - J * (n> 1)0 1 + x 0 ( l- x n )

X X

Q.35 Showthat Je"-e" z i dz = e x ^ 4 f e ' ^ 4 dzo o

Q . 3 7 ( a ) | l ^ , J

» - , (b) f d x

2a

Q.33 j x sin-1 1 2 a - x

2 V adx

Q.36 Jr x

2 sin 2x,sin (y.cos x )

2x-7 tdx

o l+x'Vx+x^x3 ' 0 1 + x 2 ' ^

OO 1 1

Q.38 Showthat f — — =2o x + 2 x c o s 9 + 1 n

d x

0 x2 + 2xcos9 + 1

sinBif 0 s (0,ic)

9 - 2 7i .

sin 9if 0 g k,2k)

? 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 jx2(sin2x-cos2x)

(1+sin 2x)cos2 x

dxX , s, Xr

u

Q.41 Prove that J f f (t) dtn J

VO J

Q.42 Jd 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




A

Q.45 Let f(x) = - j /n cos y dy then prove that f(x) = 2fv4 2,

- 2 f71 X

4 _ 2-x/n2.

Jt/2

Hence evaluate J s e c t dt

~ . ^ , r x. lnx , , r a x. dxQ.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)5d x

{ x A / (x-a) ( )3-x) V^P

px.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

- 1x -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 < 0Q.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 < J0

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) ^ < Ji 10+3cosx — 2 0

dx

2 + x2 " 6

Q.6 Determine a positive integer n<5, such that J e x (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 + X

2 1 + x4 1+x8

/UN 1,1 I / ! .U * l - 2 x 2x-4x 3 4x 3 -8x7 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 + tan^j + + 77 tan = — 7 cot ^77 - 2cot 2x

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

2

a 0 " dxX

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).

dxQ. 12 If f (9) = —

w de I -~ COS0 cosx

. 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 is finite and non zero. Determine the value of'c' and the limit.

f t2 dtJ 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

then find 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 = 00

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

X ~F~ T 40




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) ^ A1/n

(b) L i m1 + - 1 + - 1 + - 1+ -

l/n

(c)Lim I 1 2 3n+ + + —

n+1 n+2 4n. . . . . 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 2kx

k e N and hencesinx

71/2

prove that, j1 1 1 1£ H L ^ d x = : i + - + _ + ^ + +

sin x 3 5 7 . 2k-l

"J? • 2Q.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> I m ; I 1 =jxm . ( lnx)n dx = (-l )n

(m+n+1)!

n!

m, n s N.

m, n s N.n+io (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^jd t+e 2

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 dxQ.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 1Q.29 Prove that £ ("l) k k 1 —T = S ( " ^ ( i f ) ,k = o 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 V

3n'+2n-l

X

Q.2 (a) If g (x) = J c 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 '97,2 + 2 + 2 + 2 + 5]„ 1 + COS X

Q.3 (a) If ff(t)dt = x+ft 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 1 x dx . Hence or otherwise, evaluate the integral

i U-x+x 2 ; i

iJtan _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

371/2integral j [2 sinx] dx is:

(b)

(A) - n (B) 0

3,1/4 rlYf ——— 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 . -cosxe +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 jlogex

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] and0< 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 + VnFind

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 + 2Q.9 Evaluate f sin 1

W4X2+8X + 13

IT/:

Q.10 (a) Evaluate }

dx.

i t/2 9 51

cos x r xdx —5 r r ~ d x . (b) Evaluate J :

0 cos3 x + sin

3 x / 0 1 + cos a sinx

[REE2001, 3+ 5]




X

Q. 11 (a) Let f(x) = j ^ 2 - t 2 dt Then the real roots ofthe equation x2 - f ' (x) = 0are

(A)±l (B) ± -"" (C) ±- (D) 0 and 1

(h) Let T > 0 be a fixed real number. Suppose/ is a continuous function such that for all x e RT 3+3T

/ ( x + T) = /(x) . If I = f /fx; dx then the value of J ' j W dx is0 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 dx0 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 =- ^t 5 , t>0, thenf

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

vV3yis

(C)3

(D)l

[JEE 2004, (Scr.)]

[JEE 2004 (Mains), 2]

[JEE 2004 (Mains), 4]

[JEE 2005 (Scr.)]

(D)V3

feBansal Classes Definite & Indefinite Integration [15]



(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

-cosxv2 ,

+3 cosx' N

— cosxv2 j j

sin x dx.

[JEE 2005 (Scr.)]

(D)6

[JEE 2005, Mains,2]

Q15 ^W5?^T

dXisequalt0

( A ) V 2 x 4 - 2 x 2+ l + c

(B)

(C)x -2x z + l

+ CD)

V2X4-2X2+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)), for

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 - aJf (x)dx = ( f ( a ) + f ( b ) + 2f (c>)

a

i t / 2

(a) | sin xdx is equal too

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

(D)71

4V2

j f ( 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 - aB)

2(f(b)-f(a))

b - a(C)

2f(b)~f(a)

2b-a

5050 } ( I - x 5 ° R dx

Q. 17 Find the value of

J ( I - x 5° R <

(D)0

[JEE 2006, 5 marks each]

[JEE 2006,6]dx

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



ANSWER KEY

EXERCISE-I

Q.l Inl + Vl + 3cos

2 20

cos 20+ C

^ _ x + 1Q.2 -

7 C

x + x + 1

1 x j o x 3

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

-1

x - —— 7 - — In2 o 0

Q.5 2 tan-1

(x+Vx2+2x-l) + c

— I all X - / . \ - —8 4(x -1) 16

f iNx - 1

VX + V + c

Q.6 I*.e/

- - +c

Q.7 (c) - (sin 20) Incos9 + sinG ] 1

cos9 - sin9j 2

1- - /n( sec20) + 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 C

f \cosx

vcosay

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

Q.14 - In tan-, 1 , x x+ - sec2 - +tan - + c

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

Q.18 xln (lnx) - —— + clnx

8(tan x)

Q.15 V^Jl^x-2-Jl^ + arc cos Vx+c

(x2

+ l \l

+ c

Q.17

Q.19 In

/x 2 +1

9x

xe"

vl+xex j

2-31nfl+-^

l+xex

V x

+ c

Q . 2 0 - / n ( l - x4 ) + c Q.21 6t4 t2 1

— - — +1 + ~/n( l + t z ) - tan-1 1

4 2 2

Q1 + -yjcOSj

. 2 2 . +2tan"1 Jcosf -Inycosf

v 1 - ycosi

+ C where t = x1/6

+ c Q.23 -vx- ly

+ C Q24. sin"1 —sec —v.2 2 j

C

~ 1 , (4+3sin x+3cos x)Q.25 —In- —+c24 (4-3sin x-3cos x)

Q 27 1 VT+sin x-cos x

Q . 2 « 2 sin x - cos x —p=ln tanV2

F X 71 ^C

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

InV sina

Q.31 2x-3arctan

cotx+cota+^/cot2x+2cotacotx-lx sin x +cosx

+ c Q.30 — + cxc os x - s m x

' x ^tan—+1

v 2 ,+c Q.32

Vcos2x

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




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 =

Vco tx

_ 1, X 1 2X

Q.34 —lntan tan —+c2 2 4 2

Q.35 c -(x

2- l )

2

Q.36 c - ecos x

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

cx J

+k Q.38 ex J ^ + c Q.39

20>> +c

_ lnxQ.40 arcsecx—, +c

i

I 2

—Q.41 In , ' + V3tan

Vu4+u

2 + l

/r l+2uV3tr~ 1

9^7x-10-x2

V3+ c where u = 3

V 1+x

Q.42 -1 . f V51 - 1

tan t + —r= ~7=2V5 vv5t + 1

- |sin 1

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

1+x

x

Q.43 tan. 2 sin 2x

vsinx + cosxy

+ c7 6x

Q.44 4/nx+-+6tan-1( x) + j +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+lsin

^ 3+ c

-2 x ^Q.47 —-J—-+c

a-p y x- a

1 4 - - 2 15 ,Q.48

n x x 2 + — In

4x2 y l + x 8

VT+x- i

7 1 + X + 1+ c

Q.49V2

nl V2-t

- —/n2

' 1 - 0

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

Q.50acosec—2

tan-1\

v , 2 x

y

acosec—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.122n + l

f 22 ^7t

17 J

7tQ . 1 3 - ( l - l n 4 )

Q.14 4V2-4/n(V2 + l) Q.15 ^ Q.16 Q . 17 ^ = Q. 1 8 - ^ ( e2

* + l )

_ 71 16V2 _ (aln2-5a+^)

Q.22(a7t + 2b)7t

3V3Q.23

7t(7t + 3)Q.24

Q.21 i3

371+1

7t2

V2 l"arc tan arc tan-

3 3

Q 27




Q.2871

2a(a+b)

57t

3Q.29 Q.30 ^ L Q.31 (a) ^=[rc+21n(V2-l)] (b)

16 12

2 2 2 2Q.32—5-cos7ixforO<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 + 12

167t _ rr 271Q.44 (a) — 2V3 (b) i - J ^ b 2 Tt

Q.45 - / n 2 Q.4771 + 4

666

32Q.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 . lO l+ e Q.ll f(x) =x+1—x 119 119

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

if 0<x<l

2

x2

X - TQ.16 x = 2 or 4 Q.17 F(x)= \ if l<x<2

k if 2<x<3M .3 1 2

Q.18 f(x) = l +x 2 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


definite and indefinite integration

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