JEE Exam > JEE Notes > Mathematics (Maths) Main & Advanced > JEE Main Previous Year Questions (2021-2026): Definite Integral
JEE Main Previous Year Questions (2021-2026): Definite Integral
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1 JEE Main Previous Year Questions (2025): Definite Integrals Q1: Let ?? : ( ?? , 8 ) ? ?? be a twice differentiable function. If for some ?? ? ?? , ? ? ?? ?? ?? ( ???? ) ?? ?? = ???? ( ?? ) , ?? ( ?? ) = ?? and ?? ( ???? ) = ?? ?? , then ???? - ?? ' ( ?? ???? ) is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Morning Shift Ans: 112 Solution: ? 0 1 ? f ( ?? x ) d ?? = af ( x ) ?? x = t d ?? = 1 x dt 1 ?? ? 0 ?? ? ?? ( ?? ) ???? = ???? ( ?? ) ? 0 x ? f ( t ) dt = ax f ( x ) f ( x ) = a ( xf ' ( x ) + f ( x ) ) ( 1 - ?? ) ?? ( ?? ) = ?? · ?? ?? ' ( ?? ) f ' ( x ) f ( x ) = ( 1 - a ) a 1 x l ln ? f ( x ) = 1 - a a l nx + c x = 1 , f ( 1 ) = 1 ? c = 0 x = 16 , f ( 16 ) = 1 8 1 8 = ( 16 ) 1 - ?? ?? ? - 3 = 4 - 4 ?? ?? ? a = 4 f ( x ) = x - 3 4 f ' ( x ) = - 3 4 x - 7 4 ? 16 - f ' ( 1 16 ) = 16 - ( - 3 4 ( 2 - 4 ) - 7 / 4 ) = 16 + 96 = 112 Page 2 JEE Main Previous Year Questions (2025): Definite Integrals Q1: Let ?? : ( ?? , 8 ) ? ?? be a twice differentiable function. If for some ?? ? ?? , ? ? ?? ?? ?? ( ???? ) ?? ?? = ???? ( ?? ) , ?? ( ?? ) = ?? and ?? ( ???? ) = ?? ?? , then ???? - ?? ' ( ?? ???? ) is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Morning Shift Ans: 112 Solution: ? 0 1 ? f ( ?? x ) d ?? = af ( x ) ?? x = t d ?? = 1 x dt 1 ?? ? 0 ?? ? ?? ( ?? ) ???? = ???? ( ?? ) ? 0 x ? f ( t ) dt = ax f ( x ) f ( x ) = a ( xf ' ( x ) + f ( x ) ) ( 1 - ?? ) ?? ( ?? ) = ?? · ?? ?? ' ( ?? ) f ' ( x ) f ( x ) = ( 1 - a ) a 1 x l ln ? f ( x ) = 1 - a a l nx + c x = 1 , f ( 1 ) = 1 ? c = 0 x = 16 , f ( 16 ) = 1 8 1 8 = ( 16 ) 1 - ?? ?? ? - 3 = 4 - 4 ?? ?? ? a = 4 f ( x ) = x - 3 4 f ' ( x ) = - 3 4 x - 7 4 ? 16 - f ' ( 1 16 ) = 16 - ( - 3 4 ( 2 - 4 ) - 7 / 4 ) = 16 + 96 = 112 Q2: If ?????? ?? ? ?? ? ( ? ?? ?? ? ( ?? ?? + ?? ) ?? ???? ) ?? ?? = ?? ?? ?? ( ?? ?? ) ?? ?? , then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 64 Solution: 1 8 form Now ?? = ?? ?? ? 0 1 ?? ( ( 3 ?? + 5 ) ?? + 1 3 ( ?? + 1 ) | 0 1 - 1 ) = ?? ?? ? 0 8 ?? + 1 - 5 ?? + 1 - 3 ?? - 3 3 ?? ( ?? + 1 ) = ?? 8ln ? 8 - 5ln ? 5 - 3 3 = ( 8 5 ) 2 / 3 ( 64 5 ) = ?? 5e ( 8 5 ) 2 / 3 On comparing ?? = 64 Q3: If ???? ? ?? ?? ?? ? [ ?? ???? ? ?? ?? - ?? ???? | ? + [ ???????? ? ?? ] ] ???? = ?? ?? + ?? , where [ · ] denotes the greatest integer function, then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 12 Solution: = 24 ? 0 ?? 48 ? - sin ? ( 4x - ?? 12 ) + ? ?? / 48 ?? / 4 ? s i n ? ( 4x - ?? 12 ) + ? 0 ?? 6 ? [ 0 ] ???? + ? ?? / 6 ?? / 4 ? [ 2s i n ? ?? ] ???? = 24 [ ( 1 - c os ? ?? 12 ) 4 - ( - c os ? ?? 12 - 1 ) 4 ] + ?? 4 - ?? 6 = 24 ( 1 2 + ?? 12 ) = 2 ?? + 12 ?? = 12 Q4: Let [.] denote the greatest integer function. If ? ?? ?? ?? ? [ ?? ?? ?? - ?? ] ???? = ?? - ???? ?? ?? ? ?? , then ?? ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 2nd April Morning Shift Ans: 8 Page 3 JEE Main Previous Year Questions (2025): Definite Integrals Q1: Let ?? : ( ?? , 8 ) ? ?? be a twice differentiable function. If for some ?? ? ?? , ? ? ?? ?? ?? ( ???? ) ?? ?? = ???? ( ?? ) , ?? ( ?? ) = ?? and ?? ( ???? ) = ?? ?? , then ???? - ?? ' ( ?? ???? ) is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Morning Shift Ans: 112 Solution: ? 0 1 ? f ( ?? x ) d ?? = af ( x ) ?? x = t d ?? = 1 x dt 1 ?? ? 0 ?? ? ?? ( ?? ) ???? = ???? ( ?? ) ? 0 x ? f ( t ) dt = ax f ( x ) f ( x ) = a ( xf ' ( x ) + f ( x ) ) ( 1 - ?? ) ?? ( ?? ) = ?? · ?? ?? ' ( ?? ) f ' ( x ) f ( x ) = ( 1 - a ) a 1 x l ln ? f ( x ) = 1 - a a l nx + c x = 1 , f ( 1 ) = 1 ? c = 0 x = 16 , f ( 16 ) = 1 8 1 8 = ( 16 ) 1 - ?? ?? ? - 3 = 4 - 4 ?? ?? ? a = 4 f ( x ) = x - 3 4 f ' ( x ) = - 3 4 x - 7 4 ? 16 - f ' ( 1 16 ) = 16 - ( - 3 4 ( 2 - 4 ) - 7 / 4 ) = 16 + 96 = 112 Q2: If ?????? ?? ? ?? ? ( ? ?? ?? ? ( ?? ?? + ?? ) ?? ???? ) ?? ?? = ?? ?? ?? ( ?? ?? ) ?? ?? , then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 64 Solution: 1 8 form Now ?? = ?? ?? ? 0 1 ?? ( ( 3 ?? + 5 ) ?? + 1 3 ( ?? + 1 ) | 0 1 - 1 ) = ?? ?? ? 0 8 ?? + 1 - 5 ?? + 1 - 3 ?? - 3 3 ?? ( ?? + 1 ) = ?? 8ln ? 8 - 5ln ? 5 - 3 3 = ( 8 5 ) 2 / 3 ( 64 5 ) = ?? 5e ( 8 5 ) 2 / 3 On comparing ?? = 64 Q3: If ???? ? ?? ?? ?? ? [ ?? ???? ? ?? ?? - ?? ???? | ? + [ ???????? ? ?? ] ] ???? = ?? ?? + ?? , where [ · ] denotes the greatest integer function, then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 12 Solution: = 24 ? 0 ?? 48 ? - sin ? ( 4x - ?? 12 ) + ? ?? / 48 ?? / 4 ? s i n ? ( 4x - ?? 12 ) + ? 0 ?? 6 ? [ 0 ] ???? + ? ?? / 6 ?? / 4 ? [ 2s i n ? ?? ] ???? = 24 [ ( 1 - c os ? ?? 12 ) 4 - ( - c os ? ?? 12 - 1 ) 4 ] + ?? 4 - ?? 6 = 24 ( 1 2 + ?? 12 ) = 2 ?? + 12 ?? = 12 Q4: Let [.] denote the greatest integer function. If ? ?? ?? ?? ? [ ?? ?? ?? - ?? ] ???? = ?? - ???? ?? ?? ? ?? , then ?? ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 2nd April Morning Shift Ans: 8 Solution: To solve this, we start by evaluating the integral: ?? = ? 0 ?? 3 ? [ 1 ?? ?? - 1 ] ???? The greatest integer function [ · ] returns the largest integer less than or equal to the input value. Here's how we can approach the problem: Determine the function inside the integral: 1 ?? ?? - 1 = ?? 1 - ?? . Identifying the intervals: When ?? 1 - ?? = 2, which simplifies to ?? = 1 - ln ? 2, we have [ ?? 1 - ?? ] = 2. When 1 = ?? 1 - ?? < 2, simplifying gives 1 - ln ? 2 < ?? = 1, and thus [ ?? 1 - ?? ] = 1. When 0 = ?? 1 - ?? < 1, which holds for ?? > 1, thus [ ?? 1 - ?? ] = 0 from ?? = 1 to ?? = ?? 3 . Evaluate the integral on these intervals: ? 0 1 - ln ? 2 ? 2 ???? = 2 ( 1 - ln ? 2 ) ? 1 - ln ? 2 1 ? 1 ???? = 1 - ( 1 - ln ? 2 ) = ln ? 2 ? 1 ?? 3 ? 0 ???? = 0 Combine these results: ?? = 2 ( 1 - ln ? 2 ) + ln ? 2 + 0 = 2 - ln ? 2 Thus, we are given that: ?? - ln ? 2 = 2 - ln ? 2 This implies that: ?? = 2 Therefore, ?? 3 = 2 3 = 8. Q5: Let for ?? ( ?? ) = ?? ?????? ?? ? ?? + ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? , ?? ?? = ? ?? ?? / ?? ? ?? ( ?? ) ?? ?? and ?? ?? = ? ?? ?? / ?? ? ???? ( ?? ) ?? ?? . Then ?? ?? ?? + ???? ?? ?? is equal to: JEE Main 2025 (Online) 22nd January Morning Shift Options: A. 2 ?? B. 1 C. ?? D. 2 Ans: B Solution: Page 4 JEE Main Previous Year Questions (2025): Definite Integrals Q1: Let ?? : ( ?? , 8 ) ? ?? be a twice differentiable function. If for some ?? ? ?? , ? ? ?? ?? ?? ( ???? ) ?? ?? = ???? ( ?? ) , ?? ( ?? ) = ?? and ?? ( ???? ) = ?? ?? , then ???? - ?? ' ( ?? ???? ) is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Morning Shift Ans: 112 Solution: ? 0 1 ? f ( ?? x ) d ?? = af ( x ) ?? x = t d ?? = 1 x dt 1 ?? ? 0 ?? ? ?? ( ?? ) ???? = ???? ( ?? ) ? 0 x ? f ( t ) dt = ax f ( x ) f ( x ) = a ( xf ' ( x ) + f ( x ) ) ( 1 - ?? ) ?? ( ?? ) = ?? · ?? ?? ' ( ?? ) f ' ( x ) f ( x ) = ( 1 - a ) a 1 x l ln ? f ( x ) = 1 - a a l nx + c x = 1 , f ( 1 ) = 1 ? c = 0 x = 16 , f ( 16 ) = 1 8 1 8 = ( 16 ) 1 - ?? ?? ? - 3 = 4 - 4 ?? ?? ? a = 4 f ( x ) = x - 3 4 f ' ( x ) = - 3 4 x - 7 4 ? 16 - f ' ( 1 16 ) = 16 - ( - 3 4 ( 2 - 4 ) - 7 / 4 ) = 16 + 96 = 112 Q2: If ?????? ?? ? ?? ? ( ? ?? ?? ? ( ?? ?? + ?? ) ?? ???? ) ?? ?? = ?? ?? ?? ( ?? ?? ) ?? ?? , then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 64 Solution: 1 8 form Now ?? = ?? ?? ? 0 1 ?? ( ( 3 ?? + 5 ) ?? + 1 3 ( ?? + 1 ) | 0 1 - 1 ) = ?? ?? ? 0 8 ?? + 1 - 5 ?? + 1 - 3 ?? - 3 3 ?? ( ?? + 1 ) = ?? 8ln ? 8 - 5ln ? 5 - 3 3 = ( 8 5 ) 2 / 3 ( 64 5 ) = ?? 5e ( 8 5 ) 2 / 3 On comparing ?? = 64 Q3: If ???? ? ?? ?? ?? ? [ ?? ???? ? ?? ?? - ?? ???? | ? + [ ???????? ? ?? ] ] ???? = ?? ?? + ?? , where [ · ] denotes the greatest integer function, then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 12 Solution: = 24 ? 0 ?? 48 ? - sin ? ( 4x - ?? 12 ) + ? ?? / 48 ?? / 4 ? s i n ? ( 4x - ?? 12 ) + ? 0 ?? 6 ? [ 0 ] ???? + ? ?? / 6 ?? / 4 ? [ 2s i n ? ?? ] ???? = 24 [ ( 1 - c os ? ?? 12 ) 4 - ( - c os ? ?? 12 - 1 ) 4 ] + ?? 4 - ?? 6 = 24 ( 1 2 + ?? 12 ) = 2 ?? + 12 ?? = 12 Q4: Let [.] denote the greatest integer function. If ? ?? ?? ?? ? [ ?? ?? ?? - ?? ] ???? = ?? - ???? ?? ?? ? ?? , then ?? ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 2nd April Morning Shift Ans: 8 Solution: To solve this, we start by evaluating the integral: ?? = ? 0 ?? 3 ? [ 1 ?? ?? - 1 ] ???? The greatest integer function [ · ] returns the largest integer less than or equal to the input value. Here's how we can approach the problem: Determine the function inside the integral: 1 ?? ?? - 1 = ?? 1 - ?? . Identifying the intervals: When ?? 1 - ?? = 2, which simplifies to ?? = 1 - ln ? 2, we have [ ?? 1 - ?? ] = 2. When 1 = ?? 1 - ?? < 2, simplifying gives 1 - ln ? 2 < ?? = 1, and thus [ ?? 1 - ?? ] = 1. When 0 = ?? 1 - ?? < 1, which holds for ?? > 1, thus [ ?? 1 - ?? ] = 0 from ?? = 1 to ?? = ?? 3 . Evaluate the integral on these intervals: ? 0 1 - ln ? 2 ? 2 ???? = 2 ( 1 - ln ? 2 ) ? 1 - ln ? 2 1 ? 1 ???? = 1 - ( 1 - ln ? 2 ) = ln ? 2 ? 1 ?? 3 ? 0 ???? = 0 Combine these results: ?? = 2 ( 1 - ln ? 2 ) + ln ? 2 + 0 = 2 - ln ? 2 Thus, we are given that: ?? - ln ? 2 = 2 - ln ? 2 This implies that: ?? = 2 Therefore, ?? 3 = 2 3 = 8. Q5: Let for ?? ( ?? ) = ?? ?????? ?? ? ?? + ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? , ?? ?? = ? ?? ?? / ?? ? ?? ( ?? ) ?? ?? and ?? ?? = ? ?? ?? / ?? ? ???? ( ?? ) ?? ?? . Then ?? ?? ?? + ???? ?? ?? is equal to: JEE Main 2025 (Online) 22nd January Morning Shift Options: A. 2 ?? B. 1 C. ?? D. 2 Ans: B Solution: ?? ( ?? ) ? = 7 tan 8 ? ?? + 7 tan 6 ? ?? - 3 tan 4 ? ?? - 3 tan 2 ? ?? ? = 7 tan 6 ? ?? ( 1 + tan 2 ? ?? ) - 3 tan 2 ? ?? ( 1 + tan 2 ? ?? ) ? = ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) ( 1 + tan 2 ? ?? ) ? = ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ?? ?? 1 = ? ? ? ?? 4 0 ? ?? ( ?? ) ???? = ? ? ?? 4 0 ? ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ???? ?? ? = ( 7 tan 7 ? ?? 7 - 3 tan 3 ? ?? 3 ) | 0 ?? 4 = 1 - 1 = 0 ?? 2 = ? ? ? ?? 4 0 ? ???? ( ?? ) ???? = ? ? ?? 4 0 ? ?? ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ???? ?? ? = ?? ( tan 7 ? ?? - tan 3 ? ?? ) | 0 ?? 4 - ? ? ?? 4 0 ? 1 · ( tan 7 ? ?? - tan 3 ? ?? ) ???? ? = 0 - ? ? ?? 4 0 ? tan 3 ? ?? ( tan 2 ? ?? - 1 ) ( tan 2 ? ?? + 1 ) ???? = ? 0 ?? 4 ? ( tan 3 ? ?? - tan 5 ? ?? ) s e c 2 ? ???? ?? = tan 4 ? ?? 4 - tan 6 ? ?? 6 | 0 ?? 4 = 1 12 Hence 7 ?? 1 + 12 ?? 2 = 1 Q6: The value of ? ?? ?? ?? ?? ? ?? ?? ( ?? ( ( ?? ???? ?? ? ?? ) ?? + ?? ) - ?? ?? ( ( ?? ???? ?? ? ?? ) ?? + ?? ) - ?? + ?? ( ( ?? - ?? ???? ?? ? ?? ) ?? + ?? ) - ?? ) ???? is Options: A. 1 B. log ?? ? 2 C. ?? 2 D. 2 Ans: A Solution: Let ln ? x = t ? dx x = dt ?? = ? 2 4 ? ?? 1 1 + ?? 2 ?? 1 1 + ?? 2 + ?? 1 1 + ( 6 - ?? ) 2 ???? Page 5 JEE Main Previous Year Questions (2025): Definite Integrals Q1: Let ?? : ( ?? , 8 ) ? ?? be a twice differentiable function. If for some ?? ? ?? , ? ? ?? ?? ?? ( ???? ) ?? ?? = ???? ( ?? ) , ?? ( ?? ) = ?? and ?? ( ???? ) = ?? ?? , then ???? - ?? ' ( ?? ???? ) is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Morning Shift Ans: 112 Solution: ? 0 1 ? f ( ?? x ) d ?? = af ( x ) ?? x = t d ?? = 1 x dt 1 ?? ? 0 ?? ? ?? ( ?? ) ???? = ???? ( ?? ) ? 0 x ? f ( t ) dt = ax f ( x ) f ( x ) = a ( xf ' ( x ) + f ( x ) ) ( 1 - ?? ) ?? ( ?? ) = ?? · ?? ?? ' ( ?? ) f ' ( x ) f ( x ) = ( 1 - a ) a 1 x l ln ? f ( x ) = 1 - a a l nx + c x = 1 , f ( 1 ) = 1 ? c = 0 x = 16 , f ( 16 ) = 1 8 1 8 = ( 16 ) 1 - ?? ?? ? - 3 = 4 - 4 ?? ?? ? a = 4 f ( x ) = x - 3 4 f ' ( x ) = - 3 4 x - 7 4 ? 16 - f ' ( 1 16 ) = 16 - ( - 3 4 ( 2 - 4 ) - 7 / 4 ) = 16 + 96 = 112 Q2: If ?????? ?? ? ?? ? ( ? ?? ?? ? ( ?? ?? + ?? ) ?? ???? ) ?? ?? = ?? ?? ?? ( ?? ?? ) ?? ?? , then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 64 Solution: 1 8 form Now ?? = ?? ?? ? 0 1 ?? ( ( 3 ?? + 5 ) ?? + 1 3 ( ?? + 1 ) | 0 1 - 1 ) = ?? ?? ? 0 8 ?? + 1 - 5 ?? + 1 - 3 ?? - 3 3 ?? ( ?? + 1 ) = ?? 8ln ? 8 - 5ln ? 5 - 3 3 = ( 8 5 ) 2 / 3 ( 64 5 ) = ?? 5e ( 8 5 ) 2 / 3 On comparing ?? = 64 Q3: If ???? ? ?? ?? ?? ? [ ?? ???? ? ?? ?? - ?? ???? | ? + [ ???????? ? ?? ] ] ???? = ?? ?? + ?? , where [ · ] denotes the greatest integer function, then ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 29th January Evening Shift Ans: 12 Solution: = 24 ? 0 ?? 48 ? - sin ? ( 4x - ?? 12 ) + ? ?? / 48 ?? / 4 ? s i n ? ( 4x - ?? 12 ) + ? 0 ?? 6 ? [ 0 ] ???? + ? ?? / 6 ?? / 4 ? [ 2s i n ? ?? ] ???? = 24 [ ( 1 - c os ? ?? 12 ) 4 - ( - c os ? ?? 12 - 1 ) 4 ] + ?? 4 - ?? 6 = 24 ( 1 2 + ?? 12 ) = 2 ?? + 12 ?? = 12 Q4: Let [.] denote the greatest integer function. If ? ?? ?? ?? ? [ ?? ?? ?? - ?? ] ???? = ?? - ???? ?? ?? ? ?? , then ?? ?? is equal to _ _ _ _ . JEE Main 2025 (Online) 2nd April Morning Shift Ans: 8 Solution: To solve this, we start by evaluating the integral: ?? = ? 0 ?? 3 ? [ 1 ?? ?? - 1 ] ???? The greatest integer function [ · ] returns the largest integer less than or equal to the input value. Here's how we can approach the problem: Determine the function inside the integral: 1 ?? ?? - 1 = ?? 1 - ?? . Identifying the intervals: When ?? 1 - ?? = 2, which simplifies to ?? = 1 - ln ? 2, we have [ ?? 1 - ?? ] = 2. When 1 = ?? 1 - ?? < 2, simplifying gives 1 - ln ? 2 < ?? = 1, and thus [ ?? 1 - ?? ] = 1. When 0 = ?? 1 - ?? < 1, which holds for ?? > 1, thus [ ?? 1 - ?? ] = 0 from ?? = 1 to ?? = ?? 3 . Evaluate the integral on these intervals: ? 0 1 - ln ? 2 ? 2 ???? = 2 ( 1 - ln ? 2 ) ? 1 - ln ? 2 1 ? 1 ???? = 1 - ( 1 - ln ? 2 ) = ln ? 2 ? 1 ?? 3 ? 0 ???? = 0 Combine these results: ?? = 2 ( 1 - ln ? 2 ) + ln ? 2 + 0 = 2 - ln ? 2 Thus, we are given that: ?? - ln ? 2 = 2 - ln ? 2 This implies that: ?? = 2 Therefore, ?? 3 = 2 3 = 8. Q5: Let for ?? ( ?? ) = ?? ?????? ?? ? ?? + ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? - ?? ?????? ?? ? ?? , ?? ?? = ? ?? ?? / ?? ? ?? ( ?? ) ?? ?? and ?? ?? = ? ?? ?? / ?? ? ???? ( ?? ) ?? ?? . Then ?? ?? ?? + ???? ?? ?? is equal to: JEE Main 2025 (Online) 22nd January Morning Shift Options: A. 2 ?? B. 1 C. ?? D. 2 Ans: B Solution: ?? ( ?? ) ? = 7 tan 8 ? ?? + 7 tan 6 ? ?? - 3 tan 4 ? ?? - 3 tan 2 ? ?? ? = 7 tan 6 ? ?? ( 1 + tan 2 ? ?? ) - 3 tan 2 ? ?? ( 1 + tan 2 ? ?? ) ? = ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) ( 1 + tan 2 ? ?? ) ? = ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ?? ?? 1 = ? ? ? ?? 4 0 ? ?? ( ?? ) ???? = ? ? ?? 4 0 ? ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ???? ?? ? = ( 7 tan 7 ? ?? 7 - 3 tan 3 ? ?? 3 ) | 0 ?? 4 = 1 - 1 = 0 ?? 2 = ? ? ? ?? 4 0 ? ???? ( ?? ) ???? = ? ? ?? 4 0 ? ?? ( 7 tan 6 ? ?? - 3 tan 2 ? ?? ) s ec 2 ? ???? ?? ? = ?? ( tan 7 ? ?? - tan 3 ? ?? ) | 0 ?? 4 - ? ? ?? 4 0 ? 1 · ( tan 7 ? ?? - tan 3 ? ?? ) ???? ? = 0 - ? ? ?? 4 0 ? tan 3 ? ?? ( tan 2 ? ?? - 1 ) ( tan 2 ? ?? + 1 ) ???? = ? 0 ?? 4 ? ( tan 3 ? ?? - tan 5 ? ?? ) s e c 2 ? ???? ?? = tan 4 ? ?? 4 - tan 6 ? ?? 6 | 0 ?? 4 = 1 12 Hence 7 ?? 1 + 12 ?? 2 = 1 Q6: The value of ? ?? ?? ?? ?? ? ?? ?? ( ?? ( ( ?? ???? ?? ? ?? ) ?? + ?? ) - ?? ?? ( ( ?? ???? ?? ? ?? ) ?? + ?? ) - ?? + ?? ( ( ?? - ?? ???? ?? ? ?? ) ?? + ?? ) - ?? ) ???? is Options: A. 1 B. log ?? ? 2 C. ?? 2 D. 2 Ans: A Solution: Let ln ? x = t ? dx x = dt ?? = ? 2 4 ? ?? 1 1 + ?? 2 ?? 1 1 + ?? 2 + ?? 1 1 + ( 6 - ?? ) 2 ???? ?? = ? 2 4 ? 1 ?? 1 + ( 6 - ?? ) 2 1 ?? 1 + ( 6 - ?? ) 2 + ?? 1 1 + ?? 2 ???? 2 ?? = ? 2 4 ????? = ( ?? ) 2 4 = 4 - 2 = 2 ?? = 1 Q7: If ?? = ? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ?? ? ?? + ?? ?? ?? ?? ?? ? ?? ?? ?? , then ? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? ? ?? + ?? ?? ?? ?? ? ?? ?? ?? equals : JEE Main 2025 (Online) 23rd January Evening Shift Options: A. ?? 2 12 B. ?? 2 4 C. ?? 2 16 D. ?? 2 8 Ans: C Solution: For I Apply king (P - 5) and add 2 ?? = ? 0 ?? / 2 ????? = ?? 2 ? ?? = ?? 4 ?? 2 = ? 0 ?? / 2 ? ?? sin ? ?? c os ? ?? sin 4 ? ?? + c os 4 ? ?? ???? Apply king and add I 2 = ?? 4 ? 0 ?? / 2 ? tan ? x s ec 2 x dx tan 4 ? x + 1 put tan 2 ? x = t ?? 8 ? 0 8 ? dt t 2 + 1 = ?? 8 · ?? 2 = ?? 2 16 Q8: If ?? ( ?? , ?? ) = ? ?? ?? ? ?? ?? - ?? ( ?? - ?? ) ?? - ?? ???? , ?? , ?? > ?? , then ?? ( ?? , ???? ) + ?? ( ???? , ???? ) is JEE Main 2025 (Online) 24th January Morning Shift Options:Read More
FAQs on JEE Main Previous Year Questions (2021-2026): Definite Integral
| 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.
About this Document
4.94/5 Rating
Apr 18, 2026 Last updated
Related Exams
Document Description: JEE Main Previous Year Questions (2021-2026): Definite Integral for JEE 2026 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for JEE Main Previous Year Questions (2021-2026): Definite Integral have been prepared according to the JEE exam syllabus. Information about JEE Main Previous Year Questions (2021-2026): Definite Integral covers topics like and JEE Main Previous Year Questions (2021-2026): Definite Integral Example, for JEE 2026 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for JEE Main Previous Year Questions (2021-2026): Definite Integral.
Introduction of JEE Main Previous Year Questions (2021-2026): Definite Integral in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & JEE Main Previous Year Questions (2021-2026): Definite Integral in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: JEE Main Previous Year Questions (2021-2026): Definite Integral
Description
JEE Main Previous Year Questions (2021 of Maths is important for the exam. Get access to all the questions with solutions asked in past years of JEE exam. Download it from EduRev.
Information about JEE Main Previous Year Questions (2021-2026): Definite Integral
In this doc you can find the meaning of JEE Main Previous Year Questions (2021-2026): Definite Integral defined & explained in the simplest way possible. Besides explaining types of JEE Main Previous Year Questions (2021-2026): Definite Integral theory, EduRev gives you an ample number of questions to practice JEE Main Previous Year Questions (2021-2026): Definite Integral tests, examples and also practice JEE tests
Related Searches
pdf , Free, Exam, ppt, Important questions, Summary, Extra Questions, JEE Main Previous Year Questions (2021-2026): Definite Integral, JEE Main Previous Year Questions (2021-2026): Definite Integral, Viva Questions, study material, mock tests for examination, past year papers, Previous Year Questions with Solutions, Semester Notes, practice quizzes, MCQs, Objective type Questions, JEE Main Previous Year Questions (2021-2026): Definite Integral, shortcuts and tricks, Sample Paper, video lectures;