JEE Main Previous Year Questions (2016-2026): Indefinite Integrals

Q.1. The integral is equal to (where C is a constant of integration)    (2020)
(1)
(2)
(3)
(4)
Ans. (1)
Let,


Let
So,

Q.2. If  where C is a constant of integration, then the ordered pair (λ, f(θ)) is equal to    (2020)
(1) (1,1 tan θ)
(2) ( 1,1 -tan θ)
(3) (-1,1 + tan θ)
(4) (1,1 + tan  θ)
Ans. (3)
We have


Let tan θ = t ⇒sec² θdθ = dt. 
Therefore,

Hence,

Q.3. For x² ≠ nπ + 1, n∈N (the set of natural numbers), the integral
      (2019)




(where c is a constant of integration)
Ans. (3, 4)
Solution. Consider the given integral







Q.4. and f(0) = 0, then the value of f(1) is:      (2019)


(3) 1/2
(4) 1/4
Ans. (4)
Solution.
f(x) =




Q.5. Let n ≥ 2 be a natural number and 0 < θ < π/2. Then is equal to (where C is a constant of integration    (2019)



Ans. (1)
Solution.








Q.6. where C is a constant of integration, then f (x) is equal to:      (2019)
(1) - 2x³ - 1
(2) - 4x³ - 1
(3) -2x³ + 1
(4) 4x³+ 1
Ans. (2)
Solution. 

Put -4x³ = θ
⇒ -12x² dx = dθ
⇒


Then, by comparison
f(x) = -4x³ - 1

Q.7. for a suitable chosen integer m and a function A (x), where C is a constant of integration, then (A(x))^m equals:      (2019)




Ans. (1)
Solution.







Comparing both sides,



Q.8. where C is a constant of integration, then f(x) is equal to:      (2019)
(1)
(2)
(3)
(4)
Ans. (4)
Solution.





Q.9. The integral  is equal to:      
(where C is a constant of integration)    (2019)




Ans. (3)
Solution.






Q.10. The integral  is equal to: (where C is a constant of integration)      (2019)




Ans. (2)
Solution.

Q.11. 
is equal to:    
(where c is a constant of integration.)    (2019)
(1) 2x + sinx + 2 sin2x + c
(2) x + 2 sinx + 2 sin2x + c
(3) x + 2 sinx + sin2x + c
(4) 2x + sinx + sin2x + c
Ans. (3)
Solution.


[ ∵ sin 2x = 2 sin x cos x and sin 3x = 3 sin x - 4 sin³x]


Q.12. where C is a constant of integration, then the function f(x) is equal to:      (2019)




Ans. (4)
Solution.




Q.13. The integral ∫ sec^2/3 x cosec^4/3 xdx is equal to:
(1) -3 tan^-1/3 x + C

(3) -3 cot^-1/3 x + C
(4) 3 tan^-1/3 x + C
(Here C is a constant of integration)      (2019)
Ans. (1)
Solution.




Q.14. If ∫e^{sec x} (sec x tan x f(x) + (sec x tan x + sec² x)) dx = e^secx f(x) + C, then a possible choice of f(x) is:      (2019)




Ans. (1)
Solution.


Q.15. where C is a constant of integration, then:      (2019)
(1) A = 1/54 and f(x) = 3 (x - 1)
(2) A = 1/81 and f(x) = 3 (x - 1)
(3) A = 1/27 and f(x) = 9 (x - 1)
(4) A = 1/54 and f(x) = 9 (x - 1)²
Ans. (1)
Solution.

Let (x - 1)² = 9 tan² θ    ....(1)

After differentiating equation ...(1), we get
2 (x - 1) dx = 18 tan θ sec²θ dθ




we get: A = 1/54 and f(x) = 3 (x - 1)

Q.16. If  where c is a constant of integration, then g(-1) is equal to:      (2019)
(1) -1
(2) 1
(3)
(4)
Ans. (3)
Solution.



Q.17. The integral  is equal to:      (2019)
(Here C is a constant of integration)




Ans. (3)
Solution.




Q.18. Let α ∈ (0, π/2) be fixed. If the integral  A(x) cos2α+B(x) sin2α+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively:      (2019)
(1)  x + α and log_e|sin(x + α)|
(2) x - α and log_e|sin(x - α)|
(3) x - α and log_e |cos(x - α)|
(4) x + α and log_e |sin(x - a)|
Ans. (2)
Solution.



Q.19. The integral is equal to:    (2018)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.





Q.20. If f  = 2x + 1, (x ∈ R - {1, -2}), then ∫ f(x)dx is equal to:  (where C is a constant of integration)     (2018)
(1) 12 log_e |1 - x| - 3x + C
(2) - 12 log_e |1 - x| + 3x + C
(3) - 12 log_e |1 - x| - 3x + C
(4) 12 log_e |1 - x| + 3x + C
Ans. (3)
Solution.


= 3 {-4ℓn|1-x - x| + C = -12ℓn |1-x| - 3x + C

Q.21. If f(x) =  dt then:    (2018)
(1) f''' (x) - f''(x) = cosx - 2x sinx
(2) f'''(x) + f''(x) - f'(x) = cosx
(3) f'''(x) + f''(x) = sinx
(4) f'''(x) + f'(x) = cosx - 2x sinx
Ans. (4)
Solution.



Q.22. If   (C is a constant of integration), then the ordered pair (K, A) is equal to    (2018)
(1) (2, 1)
(2) (2, 3)
(3) (-2, 1)
(4) (-2, 3)
Ans. (2)
Solution.
I =
=

 

Q.23. Let I_n = ∫tanⁿ xdx,(n > 1). If I₄ +I₆= a tan⁵ x + bx⁵ + C, where C is a constant of integration, then the ordered pair (a, b) is equal to    (2017)
(1) (-1/5 , 0)
(2) (-1/5 , 1)
(3) (1/5, 0)
(4) (1/5, -1)
Ans. (3)
Solution.


Let tanx = t
sec²x dx = dt



Q.24. The integral   is equal to:
(where C is a constant of integration)    (2017)
(1)
(2)
(3)
(4)
Ans. (1)
Solution.




Q.25. If  and  then the ordered pair (A,B) is equal to :(where c is a constant of integration)    (2017)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.





Q.26. The integral dx is equal to:    (2016)
(1)
(2)
(3)
(4)  
Ans. (2)
Solution.
 
Dividing numerator and denominator by x¹⁵ we get,



Q.27. If  , where k is a constant of integration, then A + B + C equals    (2016)
(1) 15/5
(2) 21/5
(3) 7/10
(4) 27/10
Ans. (1)
Solution.

tan x = t


Q.28. The integral  is equal to (where C is a constant of integration)    (2016)
(1)
(2)
(3)
(4)
Ans. (2)
Solution.