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 (Î», f(Î¸)) is equal to (2020) (1) (1,1 tan Î¸) (2) ( 1,1 -tan Î¸) (3) (-1,1 + tan Î¸) (4) (1,1 + tan Î¸) Ans. (3) We have

Let tan Î¸ = t â‡’sec^{2}Î¸dÎ¸ = dt. Therefore, Hence,

Q.3. For x^{2} â‰ nÏ€ + 1, nâˆˆN (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 â‰¥ 2 be a natural number and 0 < Î¸ < Ï€/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) - 2x^{3} - 1 (2) - 4x^{3} - 1 (3) -2x^{3} + 1 (4) 4x^{3}+ 1 Ans. (2) Solution.

Put -4x^{3} = Î¸ â‡’ -12x^{2} dx = dÎ¸ â‡’

Then, by comparison f(x) = -4x^{3} - 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.

[ âˆµ sin 2x = 2 sin x cos x and sin 3x = 3 sin x - 4 sin^{3}x]

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 âˆ« 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 âˆ«e^{sec x} (sec x tan x f(x) + (sec x tan x + sec^{2} x)) dx = e^{sec}^{x} 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 tan^{2} Î¸ ....(1)

After differentiating equation ...(1), we get 2 (x - 1) dx = 18 tan Î¸ sec^{2}Î¸ dÎ¸

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 Î± âˆˆ (0, Ï€/2) be fixed. If the integral A(x) cos2Î±+B(x) sin2Î±+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively: (2019) (1) x + Î± and log_{e}|sin(x + Î±)| (2)x - Î± and log_{e}|sin(x - Î±)| (3)x - Î± and log_{e} |cos(x - Î±)| (4)x + Î± 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 âˆˆ R âˆ’ {1, âˆ’2}), then âˆ« f(x)dx is equal to: (where C is a constant of integration) (2018) (1) 12 log_{e} |1 - x| - 3x + C (2) â€“ 12 log_{e} |1 â€“ x| + 3x + C (3) â€“ 12 log_{e} |1 â€“ x| - 3x + C (4) 12 log_{e} |1 â€“ x| + 3x + C Ans. (3) Solution.

= 3 {â€“4â„“n|1â€“x â€“ x| + C = â€“12â„“n |1â€“x| â€“ 3x + C

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) (â€“2, 1) (4) (â€“2, 3) Ans. (2) Solution. I = =

Q.23. Let I_{n} = âˆ«tan^{n} xdx,(n > 1). If I_{4} +I_{6}= a tan^{5} x + bx^{5} + 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 sec^{2}x 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 x^{15} 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.

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