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JEE Mains Previous Year Questions 
(2021-2024): Definite Integration 
2024 
Q1 - 2024 (01 Feb Shift 1) 
The value of the integral ?
0
?? 4
?
?????? sin
4
? ( 2?? )+cos
4
? ( 2?? )
 equals : 
(1) 
v2?? 2
8
 
(2) 
v2?? 2
16
 
(3) 
v2?? 2
32
 
(4) 
v2?? 2
64
 
Q2 - 2024 (01 Feb Shift 1) 
If ?
-?? /2
?? /2
?
8v2cos??????? ( 1+?? sin??? )( 1+sin
4
??? )
= ???? + ?? log
?? ? ( 3 + 2v2) , where ?? ,?? are integers, then ?? 2
+ ?? 2
 
equals 
Q3 - 2024 (01 Feb Shift 2) 
The value of ?
0
1
?( 2?? 3
- 3?? 2
- ?? + 1)
1
3
???? is equal to: 
(1) 0 
(2) 1 
(3) 2 
(4) -1 
Q4 - 2024 (01 Feb Shift 2) 
If ?
0
?? 3
?cos
4
??????? = ???? + ?? v3, where ?? and ?? are rational numbers, then 9?? + 8?? is equal to 
: 
(1) 2 
Page 2


JEE Mains Previous Year Questions 
(2021-2024): Definite Integration 
2024 
Q1 - 2024 (01 Feb Shift 1) 
The value of the integral ?
0
?? 4
?
?????? sin
4
? ( 2?? )+cos
4
? ( 2?? )
 equals : 
(1) 
v2?? 2
8
 
(2) 
v2?? 2
16
 
(3) 
v2?? 2
32
 
(4) 
v2?? 2
64
 
Q2 - 2024 (01 Feb Shift 1) 
If ?
-?? /2
?? /2
?
8v2cos??????? ( 1+?? sin??? )( 1+sin
4
??? )
= ???? + ?? log
?? ? ( 3 + 2v2) , where ?? ,?? are integers, then ?? 2
+ ?? 2
 
equals 
Q3 - 2024 (01 Feb Shift 2) 
The value of ?
0
1
?( 2?? 3
- 3?? 2
- ?? + 1)
1
3
???? is equal to: 
(1) 0 
(2) 1 
(3) 2 
(4) -1 
Q4 - 2024 (01 Feb Shift 2) 
If ?
0
?? 3
?cos
4
??????? = ???? + ?? v3, where ?? and ?? are rational numbers, then 9?? + 8?? is equal to 
: 
(1) 2 
(2) 1 
(3) 3 
(4) 
3
2
 
Q5 - 2024 (01 Feb Shift 2) 
Let ?? :( 0,8)? ?? and ?? ( ?? )= ?
0
?? ????? ( ?? )???? . If ?? ( ?? 2
)= x
4
+ x
5
, then ?
r=1
12
?f( r
2
) is equal to : 
Q6 - 2024 (27 Jan Shift 1) 
If ( ?? ,?? ) be the orthocentre of the triangle whose vertices are ( 1,2),( 2,3) and ( 3,1) , and 
?? 1
= ?
a
b
?xsin ? ( 4x- x
2
)dx,I
2
= ?
a
b
?sin ? ( 4x- x
2
)dx, then 36
I
1
I
2
 is equal to : 
(1) 72 
(2) 88 
(3) 80 
(4) 66 
Q7 - 2024 (27 Jan Shift 1) 
If ?
0
1
?
1
v3+?? +v1+?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? ,?? are rational numbers, then 2?? + 3?? -
4?? is equal to : 
(1) 4 
(2) 10 
(3) 7 
(4) 8 
Q8 - 2024 (27 Jan Shift 2) 
For 0 < a < 1, the value of the integral ?
0
?? ?
????
1-2?? cos??? +?? 2
 is : 
(1) 
?? 2
?? +?? 2
 
(2) 
?? 2
?? -?? 2
 
(3) 
?? 1-?? 2
 
Page 3


JEE Mains Previous Year Questions 
(2021-2024): Definite Integration 
2024 
Q1 - 2024 (01 Feb Shift 1) 
The value of the integral ?
0
?? 4
?
?????? sin
4
? ( 2?? )+cos
4
? ( 2?? )
 equals : 
(1) 
v2?? 2
8
 
(2) 
v2?? 2
16
 
(3) 
v2?? 2
32
 
(4) 
v2?? 2
64
 
Q2 - 2024 (01 Feb Shift 1) 
If ?
-?? /2
?? /2
?
8v2cos??????? ( 1+?? sin??? )( 1+sin
4
??? )
= ???? + ?? log
?? ? ( 3 + 2v2) , where ?? ,?? are integers, then ?? 2
+ ?? 2
 
equals 
Q3 - 2024 (01 Feb Shift 2) 
The value of ?
0
1
?( 2?? 3
- 3?? 2
- ?? + 1)
1
3
???? is equal to: 
(1) 0 
(2) 1 
(3) 2 
(4) -1 
Q4 - 2024 (01 Feb Shift 2) 
If ?
0
?? 3
?cos
4
??????? = ???? + ?? v3, where ?? and ?? are rational numbers, then 9?? + 8?? is equal to 
: 
(1) 2 
(2) 1 
(3) 3 
(4) 
3
2
 
Q5 - 2024 (01 Feb Shift 2) 
Let ?? :( 0,8)? ?? and ?? ( ?? )= ?
0
?? ????? ( ?? )???? . If ?? ( ?? 2
)= x
4
+ x
5
, then ?
r=1
12
?f( r
2
) is equal to : 
Q6 - 2024 (27 Jan Shift 1) 
If ( ?? ,?? ) be the orthocentre of the triangle whose vertices are ( 1,2),( 2,3) and ( 3,1) , and 
?? 1
= ?
a
b
?xsin ? ( 4x- x
2
)dx,I
2
= ?
a
b
?sin ? ( 4x- x
2
)dx, then 36
I
1
I
2
 is equal to : 
(1) 72 
(2) 88 
(3) 80 
(4) 66 
Q7 - 2024 (27 Jan Shift 1) 
If ?
0
1
?
1
v3+?? +v1+?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? ,?? are rational numbers, then 2?? + 3?? -
4?? is equal to : 
(1) 4 
(2) 10 
(3) 7 
(4) 8 
Q8 - 2024 (27 Jan Shift 2) 
For 0 < a < 1, the value of the integral ?
0
?? ?
????
1-2?? cos??? +?? 2
 is : 
(1) 
?? 2
?? +?? 2
 
(2) 
?? 2
?? -?? 2
 
(3) 
?? 1-?? 2
 
(4) 
?? 1+?? 2
 
Q9 - 2024 (27 Jan Shift 2) 
Let ?? ( ?? )= ?
0
?? ??? ( ?? )log
?? ?(
1-?? 1+?? )???? , where ?? is a continuous odd function. 
If ?
-?? /2
?? /2
?( ?? ( ?? )+
?? 2
cos??? 1+?? ?? )???? = (
?? ?? )
2
- ?? , then ?? is equal to 
Q10 - 2024 (27 Jan Shift 2) 
If the value of the integral 
?
-
?? 2
?? 2
?(
?? 2
cos??? 1+?? ?? +
1+sin
2
??? 1+?? sin??? 2033
)???? =
?? 4
( ?? + ?? )- 2, 
then the value of a is 
(1) 3 
(2) -
3
2
 
(3) 2 
(4) 
3
2
 
Q11 - 2024 (29 Jan Shift 2) 
If ? ?? 6
?? 3
?v1- sin ?2?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? and ?? are rational numbers, then 
3?? + 4?? - ?? is equal to 
Q12 - 2024 (30 Jan Shift 1) 
The value of lim
?? ?8
??
?? =1
?? ?
?? 3
( ?? 2
+?? 2
)( ?? 2
+3?? 2
)
 is: 
(1) 
( 2v3+3)?? 24
 
(2) 
13?? 8( 4v3+3)
 
(3) 
13( 2v3-3)?? 8
 
(4) 
?? 
8( 2v3+3)
 
Page 4


JEE Mains Previous Year Questions 
(2021-2024): Definite Integration 
2024 
Q1 - 2024 (01 Feb Shift 1) 
The value of the integral ?
0
?? 4
?
?????? sin
4
? ( 2?? )+cos
4
? ( 2?? )
 equals : 
(1) 
v2?? 2
8
 
(2) 
v2?? 2
16
 
(3) 
v2?? 2
32
 
(4) 
v2?? 2
64
 
Q2 - 2024 (01 Feb Shift 1) 
If ?
-?? /2
?? /2
?
8v2cos??????? ( 1+?? sin??? )( 1+sin
4
??? )
= ???? + ?? log
?? ? ( 3 + 2v2) , where ?? ,?? are integers, then ?? 2
+ ?? 2
 
equals 
Q3 - 2024 (01 Feb Shift 2) 
The value of ?
0
1
?( 2?? 3
- 3?? 2
- ?? + 1)
1
3
???? is equal to: 
(1) 0 
(2) 1 
(3) 2 
(4) -1 
Q4 - 2024 (01 Feb Shift 2) 
If ?
0
?? 3
?cos
4
??????? = ???? + ?? v3, where ?? and ?? are rational numbers, then 9?? + 8?? is equal to 
: 
(1) 2 
(2) 1 
(3) 3 
(4) 
3
2
 
Q5 - 2024 (01 Feb Shift 2) 
Let ?? :( 0,8)? ?? and ?? ( ?? )= ?
0
?? ????? ( ?? )???? . If ?? ( ?? 2
)= x
4
+ x
5
, then ?
r=1
12
?f( r
2
) is equal to : 
Q6 - 2024 (27 Jan Shift 1) 
If ( ?? ,?? ) be the orthocentre of the triangle whose vertices are ( 1,2),( 2,3) and ( 3,1) , and 
?? 1
= ?
a
b
?xsin ? ( 4x- x
2
)dx,I
2
= ?
a
b
?sin ? ( 4x- x
2
)dx, then 36
I
1
I
2
 is equal to : 
(1) 72 
(2) 88 
(3) 80 
(4) 66 
Q7 - 2024 (27 Jan Shift 1) 
If ?
0
1
?
1
v3+?? +v1+?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? ,?? are rational numbers, then 2?? + 3?? -
4?? is equal to : 
(1) 4 
(2) 10 
(3) 7 
(4) 8 
Q8 - 2024 (27 Jan Shift 2) 
For 0 < a < 1, the value of the integral ?
0
?? ?
????
1-2?? cos??? +?? 2
 is : 
(1) 
?? 2
?? +?? 2
 
(2) 
?? 2
?? -?? 2
 
(3) 
?? 1-?? 2
 
(4) 
?? 1+?? 2
 
Q9 - 2024 (27 Jan Shift 2) 
Let ?? ( ?? )= ?
0
?? ??? ( ?? )log
?? ?(
1-?? 1+?? )???? , where ?? is a continuous odd function. 
If ?
-?? /2
?? /2
?( ?? ( ?? )+
?? 2
cos??? 1+?? ?? )???? = (
?? ?? )
2
- ?? , then ?? is equal to 
Q10 - 2024 (27 Jan Shift 2) 
If the value of the integral 
?
-
?? 2
?? 2
?(
?? 2
cos??? 1+?? ?? +
1+sin
2
??? 1+?? sin??? 2033
)???? =
?? 4
( ?? + ?? )- 2, 
then the value of a is 
(1) 3 
(2) -
3
2
 
(3) 2 
(4) 
3
2
 
Q11 - 2024 (29 Jan Shift 2) 
If ? ?? 6
?? 3
?v1- sin ?2?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? and ?? are rational numbers, then 
3?? + 4?? - ?? is equal to 
Q12 - 2024 (30 Jan Shift 1) 
The value of lim
?? ?8
??
?? =1
?? ?
?? 3
( ?? 2
+?? 2
)( ?? 2
+3?? 2
)
 is: 
(1) 
( 2v3+3)?? 24
 
(2) 
13?? 8( 4v3+3)
 
(3) 
13( 2v3-3)?? 8
 
(4) 
?? 
8( 2v3+3)
 
 
Q13 - 2024 (30 Jan Shift 1) 
The value 9?
0
9
?[v
10?? ?? +1
]dx, where [?? ] denotes the greatest integer less than or equal to ?? , 
is 
Q14 - 2024 (30 Jan Shift 2) 
Let ?? = ?? ( ?? ) be a thrice differentiable function in ( -5,5) . Let the tangents to the curve 
?? = ?? ( ?? ) at ( 1,f( 1)) and ( 3,f( 3)) make angles 
?? 6
 and 
?? 4
, respectively with positive ?? -axis. 
If 
27?
1
3
?( ( ?? '
( ?? ))
2
+ 1)?? ''
( ?? )???? = ?? + ?? v3? where ??? ,??? are integers, then the value of ?? + ?? 
equals 
(1) -14 
(2) 26 
(3) -16 
(4) 36 
Q15 - 2024 (30 Jan Shift 2) 
Let ?? :?? ? ?? be defined ?? ( ?? )= ?? ?? 2?? + ?? ?? ?? + ???? . If ?? ( 0)= -1,?? '
( log
?? ?2)= 21 and 
?
0
log
4
?4
?( ?? ( ?? )- ???? )???? =
39
2
, then the value of |?? + ?? + ?? | equals: 
(1) 16 
(2) 10 
(3) 12 
(4) 8 
Q16 - 2024 (31 Jan Shift 1) 
If the integral 525?
0
?? 2
?sin ?2?? cos
11
2
??? ( 1+ cos
5
2
??? )
1
2
???? is equal to ( ?? v2 - 64) , then ?? is equal 
to 
Q17 - 2024 (31 Jan Shift 1) 
Page 5


JEE Mains Previous Year Questions 
(2021-2024): Definite Integration 
2024 
Q1 - 2024 (01 Feb Shift 1) 
The value of the integral ?
0
?? 4
?
?????? sin
4
? ( 2?? )+cos
4
? ( 2?? )
 equals : 
(1) 
v2?? 2
8
 
(2) 
v2?? 2
16
 
(3) 
v2?? 2
32
 
(4) 
v2?? 2
64
 
Q2 - 2024 (01 Feb Shift 1) 
If ?
-?? /2
?? /2
?
8v2cos??????? ( 1+?? sin??? )( 1+sin
4
??? )
= ???? + ?? log
?? ? ( 3 + 2v2) , where ?? ,?? are integers, then ?? 2
+ ?? 2
 
equals 
Q3 - 2024 (01 Feb Shift 2) 
The value of ?
0
1
?( 2?? 3
- 3?? 2
- ?? + 1)
1
3
???? is equal to: 
(1) 0 
(2) 1 
(3) 2 
(4) -1 
Q4 - 2024 (01 Feb Shift 2) 
If ?
0
?? 3
?cos
4
??????? = ???? + ?? v3, where ?? and ?? are rational numbers, then 9?? + 8?? is equal to 
: 
(1) 2 
(2) 1 
(3) 3 
(4) 
3
2
 
Q5 - 2024 (01 Feb Shift 2) 
Let ?? :( 0,8)? ?? and ?? ( ?? )= ?
0
?? ????? ( ?? )???? . If ?? ( ?? 2
)= x
4
+ x
5
, then ?
r=1
12
?f( r
2
) is equal to : 
Q6 - 2024 (27 Jan Shift 1) 
If ( ?? ,?? ) be the orthocentre of the triangle whose vertices are ( 1,2),( 2,3) and ( 3,1) , and 
?? 1
= ?
a
b
?xsin ? ( 4x- x
2
)dx,I
2
= ?
a
b
?sin ? ( 4x- x
2
)dx, then 36
I
1
I
2
 is equal to : 
(1) 72 
(2) 88 
(3) 80 
(4) 66 
Q7 - 2024 (27 Jan Shift 1) 
If ?
0
1
?
1
v3+?? +v1+?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? ,?? are rational numbers, then 2?? + 3?? -
4?? is equal to : 
(1) 4 
(2) 10 
(3) 7 
(4) 8 
Q8 - 2024 (27 Jan Shift 2) 
For 0 < a < 1, the value of the integral ?
0
?? ?
????
1-2?? cos??? +?? 2
 is : 
(1) 
?? 2
?? +?? 2
 
(2) 
?? 2
?? -?? 2
 
(3) 
?? 1-?? 2
 
(4) 
?? 1+?? 2
 
Q9 - 2024 (27 Jan Shift 2) 
Let ?? ( ?? )= ?
0
?? ??? ( ?? )log
?? ?(
1-?? 1+?? )???? , where ?? is a continuous odd function. 
If ?
-?? /2
?? /2
?( ?? ( ?? )+
?? 2
cos??? 1+?? ?? )???? = (
?? ?? )
2
- ?? , then ?? is equal to 
Q10 - 2024 (27 Jan Shift 2) 
If the value of the integral 
?
-
?? 2
?? 2
?(
?? 2
cos??? 1+?? ?? +
1+sin
2
??? 1+?? sin??? 2033
)???? =
?? 4
( ?? + ?? )- 2, 
then the value of a is 
(1) 3 
(2) -
3
2
 
(3) 2 
(4) 
3
2
 
Q11 - 2024 (29 Jan Shift 2) 
If ? ?? 6
?? 3
?v1- sin ?2?? ???? = ?? + ?? v2+ ?? v3, where ?? ,?? and ?? are rational numbers, then 
3?? + 4?? - ?? is equal to 
Q12 - 2024 (30 Jan Shift 1) 
The value of lim
?? ?8
??
?? =1
?? ?
?? 3
( ?? 2
+?? 2
)( ?? 2
+3?? 2
)
 is: 
(1) 
( 2v3+3)?? 24
 
(2) 
13?? 8( 4v3+3)
 
(3) 
13( 2v3-3)?? 8
 
(4) 
?? 
8( 2v3+3)
 
 
Q13 - 2024 (30 Jan Shift 1) 
The value 9?
0
9
?[v
10?? ?? +1
]dx, where [?? ] denotes the greatest integer less than or equal to ?? , 
is 
Q14 - 2024 (30 Jan Shift 2) 
Let ?? = ?? ( ?? ) be a thrice differentiable function in ( -5,5) . Let the tangents to the curve 
?? = ?? ( ?? ) at ( 1,f( 1)) and ( 3,f( 3)) make angles 
?? 6
 and 
?? 4
, respectively with positive ?? -axis. 
If 
27?
1
3
?( ( ?? '
( ?? ))
2
+ 1)?? ''
( ?? )???? = ?? + ?? v3? where ??? ,??? are integers, then the value of ?? + ?? 
equals 
(1) -14 
(2) 26 
(3) -16 
(4) 36 
Q15 - 2024 (30 Jan Shift 2) 
Let ?? :?? ? ?? be defined ?? ( ?? )= ?? ?? 2?? + ?? ?? ?? + ???? . If ?? ( 0)= -1,?? '
( log
?? ?2)= 21 and 
?
0
log
4
?4
?( ?? ( ?? )- ???? )???? =
39
2
, then the value of |?? + ?? + ?? | equals: 
(1) 16 
(2) 10 
(3) 12 
(4) 8 
Q16 - 2024 (31 Jan Shift 1) 
If the integral 525?
0
?? 2
?sin ?2?? cos
11
2
??? ( 1+ cos
5
2
??? )
1
2
???? is equal to ( ?? v2 - 64) , then ?? is equal 
to 
Q17 - 2024 (31 Jan Shift 1) 
Let ?? = ( -1,8) and ?? :?? ? R be defined as 
?? ( ?? )= ?
-1
?? ?( ?? ?? - 1)
11
( 2?? - 1)
5
( ?? - 2)
7
( ?? - 3)
12
( 2?? - 10)
61
???? 
Let ?? = Sum of square of the values of ?? , where ?? ( ?? ) attains local maxima on ?? . and ?? = 
Sum of the values of ?? , where ?? ( ?? ) attains local minima on ?? . Then, the value of ?? 2
+ 2?? 
is 
Q18 - 2024 (31 Jan Shift 1) 
Let f:R ? R be a function defined by ?? ( ?? )=
4
?? 4
?? +2
 and 
?? = ?
?? ( ?? )
?? ( 1-?? )
??? sin
4
? ( ?? ( 1 - ?? ))???? 
?? = ?
?? ( ?? )
?? ( 1-?? )
?sin
4
? ( ?? ( 1 - ?? ))???? ;?? ?
1
2
. If 
?? M = ?? N,?? ,?? ? N, then the least value of ?? 2
+ ?? 2
 is equal to 
Q19 - 2024 (31 Jan Shift 2) 
Let ?? ,?? :( 0,8)? ?? be two functions defined by ?? ( ?? )= ?
-?? ?? ?( |?? | - ?? 2
)?? -?? 2
???? and ?? ( ?? )=
?
0
?? 2
??? 1/2
?? -?? ???? . Then the value of ( ?? ( vlog
?? ?9)+ ?? ( vlog
?? ?9)) is equal to 
(1) 6 
(2) 9 
(3) 8 
(4) 10 
Q20 - 2024 (31 Jan Shift 2) 
|
120
?? 3
?
0
?? ?
?? 2
sin??? cos??? sin
4
??? +cos
4
??? ???? | is equal to

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FAQs on JEE Mains Previous Year Questions (2021-2024): Definite Integral - Mathematics (Maths) for JEE Main & Advanced

1. What is the importance of understanding integrals in JEE mains exam?

Ans. Integrals are a crucial topic in the JEE mains exam as they are extensively used in solving problems related to calculus, area under curves, and other mathematical concepts. A thorough understanding of integrals is necessary to score well in the exam.

2. How can one improve their skills in solving integral-related questions for JEE mains?

Ans. Practice is key to improving skills in solving integral-related questions for JEE mains. Students should solve a variety of problems, seek help from teachers or online resources, and participate in mock tests to enhance their understanding and problem-solving abilities.

3. What are some common types of integral problems that are frequently asked in JEE mains exam?

Ans. Some common types of integral problems that are frequently asked in JEE mains exam include definite integrals, indefinite integrals, integration by substitution, integration by parts, and applications of integrals such as finding areas under curves and volumes of solids.

4. How can understanding integrals help in real-life applications beyond the JEE mains exam?

Ans. Understanding integrals can help in various real-life applications such as calculating areas of irregular shapes, determining volumes of objects, analyzing rates of change in physical phenomena, and solving optimization problems in fields like engineering, physics, economics, and more.

5. Are there any tips or strategies to effectively tackle integral problems in the JEE mains exam?

Ans. Some tips and strategies to effectively tackle integral problems in the JEE mains exam include practicing regularly, understanding the concepts thoroughly, breaking down complex problems into smaller steps, utilizing formulas and techniques effectively, and managing time efficiently during the exam.

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