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The document contains a series of mathematical problems and solutions related to indefinite integrals, primarily aimed at students preparing for the JEE Main examination. It includes various integration techniques, properties, and specific integral evaluations, along with constants of integration. The content is structured in a question-answer format, providing a comprehensive review of integral calculus concepts.
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Save Integration For Later Indefinite Integrals Ps {EE MAIN OBJECTIVE war IEMATICS ip
= MATHEMATICS ig
it Cte ne
+1
b) tan! (et —
I Ax 2 :
J iog(et* -e* +1) +e
a5
) nme te") te A) tan "Ce" 6) pg
if {1
| ex gyn g(x) +e.gt1)=0, then &| ,) is equal to
48, [ove 126 June 2022, § 4
Bn
fi-1) V3+1) x
a) we Frat }*5 b) log, ipa +z) log,
3
Pijx-t
—,|—de =
ba! Laver
a) cosh“! x—sec™
R
6
1 b) cosh x+sec"x+¢
xte
) sith x-sectxte 4) sinh x-cosech"x+¢
1
; tt . 5
50. The integral J Yeaicaray® ' caval to (Where, C is a constant of integration)
(31 Aug. 2021, s.1
1/4 s/4 14
3(x+2 3 a 4 =) 4(x-1¥"4
at =|——]| +C a =\—
a) 3242) 2) (= Vs\za2) "© (33) re
dx
Si. leer ern
(2) +3)
8 18 8 8
8(x-2 8(x-2 8( x42 3(x42
8(x-2)" B(xn2\" 8( x42 3(x42
a) A aan 323) ao!) (3) in teal a
1
r=
52. Il Peaesaare
6(2x+1)'" 6(2x+1)"
2) (241) ss » $24) soir!)
Sx* +738
53, IE £0)= [a(x 20), and /(0) = 0, then the value of (1) is [9 Jan. 2019, S
(7 +142x7)P
1 1 1
)-> ba wt Ns
Sx4+4x5
54, If t= | preand (0) = 0 then ft) is
x+1)
a)l b) U2 SB aus
ELITE SERIES for Sr. ICON Stud ‘indefinite Integral
55. The integrat [—* © (Where, C is 4 constant of integration
WES a8 re rere ap :
112" Jan, 2019, g.1
Taste by) ———* Cg * #€ g) . +c
62x +384p oY Bay +3741 9 Get y ae aap 43x" +1)
th [2 rae
Soe eel eae a7 canal to. (whicre tna constautor integration)
er 2 ;
eae = D te
9 2S 42 +0) 6S +a? +1) +1)
Ba eee mee iy knoe o then the yalue of ‘a’ is
tanx~2
al »2 03 4) 4
cos
58. The value of le a
= ;
a) = -+log|sinx+cosx|-+c
x+log|sinx+cos af)+
¢) 2log|sinx + cosx|+x4c d) None of these
ep
3
Je
Let fa)=\= de(x20). Then, £3) — f1) is equal to [4" Sep. 2020, §
a
eee yam, ys o) 443 fd eye
62 4 m2 & 62 4 1 4
60. Let f(x)= Fog a eH amd ft0) = 0, Then fl) =
(+x? )(1+V1+,
x
8) log.(1+ V2) b) log (+ v2)—4 ©) g(t V2)+% gy tog, (W314
w A (gle
HT MEI ase olsen
62. 1 J=—
a
dx =—Tan™(3Tan x)-+constant then the maximum of asinx+beosx j
? x+5? cos? x 12
[6" Apr. 2024, S-
a) 40 b) Jar ©) V39 d Jp
LITE SERIES for Sr ICON Students} ——— a
63.
64
indefinite Integrals JEE MAIN OBJECTIVE MATHEMATICS
ne te PART]
1 oe :
alioreudaaencd aan ore . inen J{ 4 =
cos? x-+4sinxcosx-+ 4sin” x f(*) 4
a)2 b)3 4 41
_= M40 Pt kiwhere k i
If |= nena (tanx)* + Cana) +k, where & is a constant of integration, then A + B 5 ¢
equal
[2016
16 Pe a 21 :
a) 5 10 °c) 10 d) TE
65.
66.
67.
69.
70.
rf ee a(tan? x+b)Vtanx +c where ‘c’ is constant of integration then 25a? + 25 =
cos’ asin 2x
a) 3 b) 12 0) 6 48
(na - cosec x+sinx (x
z a ee ;
For xe(-4). y@) eT and : y(x) =0 then (4) is equal to
~ [29 Jan. 2024, S.t]
Qe a @ ee
do
era aeooe e600) Oflan26 +3628) Atan@ +2log,|f(0)|-+C where C is a constant of integration, then th
‘ordered pair (A, f(@)) is equal to (9% Jan. 2020, S-II
a) (1,1+ tan@) b) (11-tan8) c) (-1,1+ tan@) d) (-1,1-tan8)
— eon : 5
cos" 6(6e029 — tan 28) Atan6 —2log|/(@)|+C where °C’ is a constant of integra tion the
(A, (8) =
a) (1,1+tan) b) (-1,1-tané) c) (L.1+tand) 4) (i,1-tan)
If J (O¥ +208 —e* Ie dx = g(x)e(t*) +c, where c is a constant of integration, then g(0)
equal to [5 Sep. 2020, S-
a2 b) ce dt
Let n > 2 be a natural number and o Le ic #4indefinite integrals
(JEE MAIN OBJECTIVE MATHEMATICS IIB PART-2 9)
1. The value of the integral J ° 10
l
Feb, 2021, S-1]
where, € is a constant of integration
: 1 ‘ ‘ spt»
») Lpit-18sin?@+9sin* O~2sin’ OP + ¢ by “aI9~2eon’ 04-300 0-6cos? o}? +C
we
ve 1 xa bom 3
«) 1 Y9-2sin®@=3sin' —6sin’ OP +C 4) -L[11-18c0s? 0+9c08" 0 -2e0s" OP +C
8 18
> ! bod dQ
- vsin" @ +cos°O
|1+/14-3e0s? 20
Pn
[60820
pli 3e0s420] ql Yit3e0 261.
a! eowzo |" ay sin26— |
== Fail +sin®2)!”* +e, where C is a constant of integration, then AY\ ) is
[8 Jan. 2020, S-I]
cosxdx
73. 1) a}
sin? a(1-+sin® x)"
equal to
a)2 b) -9/8 2 4) 9/8
3 5
sin? x+cos? — ——_——
14, 1t{[7——s cos? x y= AvcosOtanx—sind + BycosO—sindcotx+C Where C is th
ysin’ xcos* xsin(x-@)
integration constant, then AB is equal to {29 Jan. 2024, S-Il
a) 4cosec(20) b) 4secO c) 2sec d) 8cosec (20)
3 2
sin’!? x-+005"? x
in(x +0)
15 dx = Avian xcos0 +sin0 ~ BYcos0 + cot x.sind + ¢
Where ‘c’ is the integration constant then A/B is
4) cod b) tan ©) sin20 a) cosecd
76, ‘The value of J ct as
(x=1)'(x-2)
5
a) Tiogts =1)+—— + 4log(x=2)+6 b) ~Mogls~1)+ 7+ 8logle =2)-he
1 1
c aan
d) = ToaLs
ELITE SERIES for Sr. ICON Students/?) —OEE--—lLlLlrl—s~——
Indefinite Integrals;5;————~
1
77. The value of the integral lee de
1 1 cS
03 2 iog|x+ t|-Geloa le? -x +14 Tan |
a
—tYJEE MAIN OBJECTIVE MATHEMATICS lip PARR
—_——- —
Bet ve
B
rol
» > + lle +1100)? +xet Tan (Bet ee
) be -loge? = 1+) +c d) None
x+l
xo+2x a
at Pe te = —In|x? —1| -—In| x +12 nf +4f4c then a+ B= __
7a, Ut fp dem inf Hn |
a)9 b) -9
ae
79, The value of [7a
a) 5 tog|x-i|-9in|x-2)+22njx—3]+¢
¢) A tog|x—1+61n)x-2)+2in|r—3]4x4C
5 2
80. The value of oa dx
a) ft 's—tran(2)+c
dran-t(x)+_Ltan- 2
©) aon (x)+ 5 Tan (Z}+¢
©) 6 d)3
b) 2 log|s+1]+9ins—2]+in|s—3)+c
d) log|x—i|=6in|2—2]+2tn|x—3]s +c
afx ax
b) 37an (Z}+270n [5}+
fn) Tau ‘wy drar'( 2 ec
: x
ee a ea de. If £(3)=5 (log, 5—log, 6), then f(4) is equal to
1 1 (25 Jan, 2023, S-1
8) 7(o819-loge17) 4) (og, 17-l0g,19) c) Jog, 19-10g,20 dog, 17 Hog, 18
tet Fe ;
82, Let f= of, if £0)=og!—Iog| then 2) =
1 im
a) {eet —1g?| » hoe! ~loat| °) hie’ log?| 4) foes”
(x+1)
q
83. Let = Jeera? 0 fim 1(2)=0, then 1(1) is equal to
e+2
% ea B(e+1) by Ss ogsfe+)
TooPe_
[8 April 2023, S-I]
e+1
+1
9 tos. (e+1) d) 5 Hoele+) [JEE MAIN OBJECTIVE MATHEMATICS IIB PART-2)}; indefinite Integrals]
att 20.1 Lt Hx)=0 then 1= (1)
84. Let (= oo
a) 1+logli+el b) I-loglt-el ¢) 1+log|l—el d) 1-logii+el
af [= :
go. ar fein ( = Jés=Acom '()+B3)+C, where Cis aconstant of integration, then the ordered
pair (A(x), BGs) can be [3" Sep. 2020, S-I1)
a) (x-1vx) b) («+ 1x) ©) (x+1,—Vx) 8) (x-1,-V)
* ta
+¢) then (A, a) =
= Alo;
86. ut fee a fe =e
oli) v(goa) of
Part-II (Numerical / Integer Type Questions)
2
dem Alog “le BTantx+C then 4 a
I= B
x
87. Evaluate le
1
88. If Shae p.Tan-!x+qTan“(x*)+C then ite
PD +1
P
89. If SO) = Li (ox+4x" + aH + 2m"), (
100. sin? x.cos*x de = Asin
101.1] Vsec2x—Idr = orlog,
7/2 (B+ Csin® x) +d- Then A ~7B + 11C =
cos2x+B+ os +0085 :)
jcos2x+ 5 +,/cos2x(1+cos Bx)
103, [aa = sin“ [A(sinx+c0sx)]+¢ then A=
¥B=sin2x
5x5 +72
104iiF $= JF presrz FO =0 If and f()=
+ constant, then B—« is equal to
(30 Jan. 2023, $-1
+e then 20+ is
102.1f J vSec2x—ldx= alog
, then the value of K is
(18 Mar. 2021, $1]
Tx +8.
105.1 ae =0 and io=% then ‘K’ is equal to
Tx ——————_
106, Let “(x)= eae ane dx Wf 10) =0 then (% | =
wl sc Where 'C’ is the constant of integration, then the valet
ies tr ry a
[8 Apr. 2024, SH
of +B +20AB is
1 (2)
dr=A/<—]| +c =
108. IF Raat x43 then SAB= __
If ate ace Tan! x+ g(x)+K , then ¢(0) =
10. why 1 ay
ie ray ot draalg(t}+bran“(3]+c tena tb=
stess gee For SH ICON Ser
= et |JEE MAIN OBJECTIVE MATHEMATICS IIB PART-2)%,
dx
112 rlop*"
al
ol saa [TO then a=
“indefinite integrais
de =alog|x~1|+blog|x? + 1|—CeyTan!(x)+C then a -b=
ar
ia.
(x= x? +41)
dja 2b 3)d 4ja
i)b 12)d 13)¢ 14)c
21i)a 22)a 23)a 24)b
31)b 32)a «-33)c 0 34)
41)b 42)b 43a 44
51)b 52)a 53)c S54)e
61)b 62)a 63)b 64)a
(Tid 72)b 73)e 74) d
81)b 82)a 83)a_-84)b
91)0 92)0 93)05 94)1
401)1 102)1 103) 0.33104) 4
411)1 112)4 113)3
5)b
15)a
25) b
35) b
45)c
55) b
65) b
75) b
85) ¢
95) 3
105) 3
6)a
16) a
26) a
36) b
48) b
56) a
66) c
76) b
86) d
96) 3
106) 5
EXERCISE - 2
7)a
iT)a
27) a
37)c
47)b
57)b
67)c
Tha
87)1
97)7
107) 7
Ba
18) b
28)a
38) ¢
48) a
58) b
68) b
78)a
88) 4
98) 1
108) 1
9) b
19)a
29)a
39)c
49)a
59) d
69) a
79a
89)3
99) 5
109) 1
10) b
20) a
30) d
40) b
50) ¢
60) b
70) ¢
80) d
90) 1
100) 0
110) 4
(Topic : Special types of Integration, Rational functions of sinx and cosx)
Part-| (Straight Objective Type Questions)
ee k=
(2x+1)Vx7 -x-2
tn ( $247)
6x43
b) ~Fgin"( a
V5 x=
-
‘_.
are
3
