Title 4-integral-calculus-1.pdf ✅

4. Integral Calculus-1 1. If ∫ dx cos3 x√2sin 2x = (tan x) A + C(tan x) B + k, where k is a constant of integration, then A + B + C equals : (A) 21 5 (B) 16 5 (C) 7 10 (D) 27 10 2. The integral ∫ dx (1+√x)√x−x 2 is equal to : (where C is a constant of integration) (A) − 2√ 1+√x 1−√x + C (B) − 2√ 1−√x 1+√x + C (C) − √ 1−√x 1+√x + C (D) 2√ 1+√x 1−√x + C 3. The integral ∫ dx (x+1) 3/4(x−2) 5/4 is equal to : (A) 4 ( x+1 x−2 ) 1/4 + C (B) 4 ( x−2 x+1 ) 1/4 + C (C) − 4 3 ( x+1 x−2 ) 1/4 + C (D) − 4 3 ( x−2 x+1 ) 1/4 + C 4. If ∫ log (t+√1+t 2) √1+t 2 dt = 1 2 (g(t)) 2 + C, where C is a constant, the g(2) is equal to : (A) 2log (2 + √5) (B) log (2 + √5) (C) 1 √5 log (2 + √5) (D) 1 2 log (2 + √5) 5. ∫ sin8 x−cos8 x (1−2sin2 xcos2 x) dx is equal to : (A) 1 2 sin 2x + c (B) − 1 2 sin 2x + c (C) − 1 2 sin x + c (D) −sin2 x + c 6. The integral ∫ xcos−1 ( 1−x 2 1+x 2 ) dx(x > 0) is equal to : (A) −x + (1 + x 2 )tan−1 x + c (B) x − (1 + x 2 )cot−1 x + c (C) − x + (1 + x 2 )cot−1 x + c (D) x − (1 + x 2 )tan−1 x + c 7. The integral ∫ sin2 xcos2 x (sin3 x+cos3 x) 2 dx is equal to : (A) 1 (1+cot3 x) + c (B) − cos3 x 3(1+sin3 x) + c (C) − 1 3(1+tan3 x) + c (D) sin3 x (1+cos3 x) + c 8. If m is a non-zero number and ∫ x 5m−1+2x 4m−1 (x 2m+xm+1) 3 dx = f(x) + c, then f(x) is : (A) x 5x 2m(x 2m+xm+1) 2 (B) x 4m 2m(x 2m+xm+1) 2 (C) 2m(x 5m+x 4m) (x 2m+xm+1) 2 (D) (x 5m−x 4m) 2m(x 2m+xm+1) 2 9. If ∫ dx x+x 7 = p(x) then, ∫ x 6 x+x 7 dx is equal to : (A) ln |x| − p(x) + c (B) ln |x| + p(x) + c (C) x − p(x) + c (D) x + p(x) + c 10. If ∫ x 2−x+1 x 2+1 e cot−1 xdx = A(x)e cot−1 x + C, then A(x) is equal to : (A) −x (B) x (C) √1 − x (D) √1 + x 11. The integral ∫ xdx 2−x 2+√2−x 2 equals : (A) log |1 + √2 + x 2| + c (B) − log |1 + √2 − x 2| + c (C) − xlog |1 − √2 − x 2| + c (D) xlog |1 − √2 + x 2| + c
12. The value of ∫ dx √x(1+√x) is : (A) 2log |√x| + C (B) 2log |1 + √x| + C (C) log |1 + √x| + C (D) None of these 13. The value of ∫ 1+cot x (x+log sin x) dx is : (A) log |x + log sin x| + C (B) log |x − log sin x| + C (C) log |x 2 + log sin x| + C (D) None of these 14. The value of ∫ log (sin x) tan x dx is : (A) [log (sin x)] 2 2 + C (B) [log (sin x)] 2 + C (C) [(sin x)] 2 2 + C (D) None of these 15. The value of ∫ x√x + x 2dx is : (A) 1 3 (x 2 + x) 3/2 − 1 4 [(x + 1 2 )√x 2 + x − 1 4 log (x + 1 2 + √x 2 + x)] + c (B) 1 3 (x 2 + x) 3/2 + 1 4 [(x + 1 2 )√x 2 + x + 1 4 log (x + 1 2 + √x 2 + x)] + c (C) 1 3 (x 2 + x) 3/2 + 1 4 [(x 2 + 1 2 )√x 2 + x − 1 4 log (x + 1 2 + √x 2 + x)] + c (D) None of these 16. The value of ∫ dx sin x⋅sin (x+α) is equal to : (A) cosec αln | sin x sin (x+α) | + C (B) cosec αln | sin (x+α) sin x | + C (C) cosec αln | sec (x+α) sec x | + C (D) cosec αln | sec x sec (x+α) | + C 17. If ∫ (sin 2x − cos 2x)dx = 1 √2 sin (2x − a) + b, then : (A) a = 5π 4 , b ∈ R (B) a = − 5π 4 , b ∈ R (C) a = π 4 , b ∈ R (D) None of these 18. The value of ∫ cos 2x cos x dx is equal to : (A) 2sin x − ln|sec x + tan x| + C (B) 2sin x − ln |sec x − tan x| + C (C) 2sin x + ln |sec x + tan x| + C (D) None of these 19. The value of ∫ 5 5 5 x ⋅ 5 5 x ⋅ 5 xdx is equal to : (A) 5 5 x (ln 5) 3 + C (B) 5 5 5 x (ln 5) 3 + C (C) 5 5 5 x (ln 5) 3 + C (D) None of these 20. The value of ∫ √tan x sin xcos x dx is equal to : (A) 2√tan x + C (B) 2√cot x + C (D) √tan x 2 + C (D) None of these 21. The value of ∫ tan3 2xsec 2xdx is equal to : (A) 1 3 sec3 2x − 1 2 sec 2x + C (B) − 1 6 sec3 2x − 1 2 sec 2x + C (C) 1 6 sec3 2x − 1 2 sec 2x + C (D) 1 3 sec3 2x + 1 2 sec 2x + C 22. The vale of ∫ cos 2x (sin x+cos x) 2 dx is equal to : (A) −1 sin x+cos x + C (B) ln |(sin x + cos x)| + C (C) ln |(sin x − cos x)| + C (D) ln (sin x + cos x) 2 + C 23. If ∫ x 13/2 ⋅ (1 + x 5/2 ) 1/2 dx = A(1 + x 5/2 ) 7/2 + B(1 + x 5/2 ) 5/2 + C(1 + x 5/2 ) 3/2 , then A, B and C are : (A) A = 4 35 , B = − 8 25 , C = 4 15 (B) A = 4 35 , B = 8 25 , C = 4 15 (C) A = − 4 35 , B = − 8 25 , C = 4 15 (D) A = 4 35 , B = − 8 25 , C = − 4 15 24. If ∫ e 3x cos 4xdx = e 3x (Asin 4x + Bcos 4x) + C then : (A) 4 A = 3 B (B) 2 A = 3 B (C) 3 A = 4 B (D) 4 A + 3 B = 1 25. The value of ∫ 1 x 2(x 4+1) 3/4 dx is equal to : (A) (1 + 1 x 4 ) 1/4 + C (B) (x 4 + 1) 1/4 + C
(C) (1 − 1 x 4 ) 1/4 + C (D) − (1 + 1 x 4 ) 1/4 + C 26. If ∫ dx x 4+x 3 = A x 2 + B x + ln | x x+1 | + C, then : (A) A = 1 2 , B = 1 (B) A = 1,B = − 1 2 (C) A = − 1 2 , B = 1 (D) None of these 27. If ∫ √ cos3 x sin11 x dx = −2(Atan−9/2 x + Btan−5/2 x) + C, then : (A) A = 1 9 , B = −1 5 (B) A = 1 9 , B = 1 5 (C) A = − 1 9 , B = 1 5 (D) None of these 28. The value of ∫ √sec x − 1dx is equal to : (A) 2ln (cos x 2 + √cos2 x 2 − 1 2 ) + C (B) ln (cos x 2 + √cos2 x 2 − 1 2 ) + C (C) − 2ln (cos x 2 + √cos2 x 2 − 1 2 ) + C (D) None of these 29. Let f ′ (x) = 3x 2 sin 1 x − xcos 1 x , if x ≠ 0, f(0) = 0 and f ( 1 π ) = 0 then : (A) f(x) is continuous at x = 0 (B) f(x) is non-derivable at x = 0 (C) f ′ (x) is continuous at x = 0 (D) f ′ (x) is non-derivable at x = 0 30. Statement 1 : If x > 0, x ≠ 1 then ∫ (logx e − (logx e) 2 )dx = xlogx e + C Statement 2 : ∫ e x (f(x) + f ′ (x))dx = e xf(x) + C and e t = x iff t = ln x (A) Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 31. The value of ∫ tan x ⋅ tan 2x ⋅ tan 3xdx is : (A) − ln|sec x| + 1 3 ln|sec 3x| − 1 2 ln|sec 2x| + C (B) ln |sec x| + 1 3 ln |sec 3x| − 1 2 ln |sec 2x| + C (C) ln |sec x| − 1 3 ln|sec 3x| + 1 2 ln |sec 2x| + C (D) None of these 32. The value of ∫ e sin x xcos3 x−sin x cos2 x dx is : (A) e sin x (x + sec x) + C (B) e sin x (x + cos x) + C (C) e sin x (x − sec x) + C (D) None of these 33. The value of ∫ √4+x 2 x 6 dx is 1 120 ( (x 2+4) b/a (x 2−6) x 5 ) + C then the value of a + b is : (A) 5 (B) 2 (C) 3 (D) None of these 34. The value of ∫ 1+xcos x x(1−x 2e 2sin x) ⋅ dx is : (A) ln |xe sin x | + 1 2 log |1 + x 2e 2sin x | + K (B) ln |xe sin x | − 1 2 log |1 − x 2e 2sin x | + K (C) ln |xe sin x | − 1 2 log |1 + xe 2sin x | + K (D) None of these 35. If f(x) = ∫ 2sin x−sin 2x x 3 dx, where x ≠ 0, then limx→0f ′ (x) has the value : (A) 0 (B) 1 (C) 2 (D) Not defined 36. The value of ∫ e √x √x (x + √x)dx is equal to : (A) 2e √x [√x − x + 1] + C (B) 2e √x [x − 2√x + 1] + C (C) 2e √x [x − √x + 1] + C (D) 2e √x (x + √x + 1) + C 37. The value of ∫ e tan θ (sec θ − sin θ)dθ is equal to : (A) −e tan θ sin θ + C (B) e tan θ sin θ + C (C) e tan θ sec θ + C (D) e tan θ cos θ + C
38. The value of ∫ 1−x 7 x(1+x 7) dx is equal to : (A) ln |x| + 2 7 ln|1 + x 7 | + C (B) ln|x| − 2 7 ln|1 − x 7 | + C (C) ln |x| − 2 7 ln |1 + x 7 | + C (D) ln |x| + 2 7 ln |1 − x 7 | + C 39. The value of ∫ (xe ln sin x − cos x)dx is equal to : (A) xcos x + C (B) sin x − xcos x + C (C) −e ln x cos x + C (D) sin x + xcos x + C 40. The value of ∫ {ln (1 + sin x) + xtan ( π 4 − x 2 )} dx is equal to : (A) xln(1 + sin x) + C (B) ln (1 + sin x) + C (C) − xln (1 + sin x) + C (D) ln (1 − sin x) + C 41. The value of ∫ √ x−1 x+1 ⋅ 1 x 2 dx is equal to : (A) sin−1 1 x + √x 2−1 x + C (B) √x 2−1 x + cos−1 1 x + C (C) sec−1 x − √x 2−1 x + C (D) tan−1 √x 2 + 1 − √x 2−1 x + C 42. Consider the following statements : S1 : The anti-derivative of every even function is an odd function. S2 : Primitive of 3x 4−1 (x 4+x+1) 2 w.r.t. x is x x 4+x+1 + C S3: ∫ 1 √sin3 xcos x dx = −2 √tan x + C S4 : The value of ∫ (√ a+x a−x − √ a−x a+x ) dx is equal to −2√a 2 − x 2 + C State, in order, whether S1, S2, S3, S4 are true or false : (A) FFTT (B) TTTT (C) FFFF (D) TFTF 43. The value of ∫ sin 2x sin4 x+cos4 x dx is equal to : (A) cot−1 (cot2 x) + C (B) − cot−1 (tan2 x) + C (C) tan−1 (tan2 x) + C (D) − tan−1 (cos 2x) + C 44. The value of ∫ dx √x−x 2 is equal to : (A) 2sin−1 √x + C (B) sin−1 (2x − 1) + C (C) C − 2cos−1 (2x − 1) (D) cos−1 2√x − x 2 + C 45. If In = ∫ cotn xdx and I0 + I1 + 2(I2 + ⋯ + I8 ) + I9 + I10 = A (u + u 2 2 + ⋯ + u 9 9 ) + C, where u = cot x and C is an arbitrary constant, then : (A) A is constant (B) A=-1 (C) A=1 (D) A is dependent on 46. Value of integral I = ∫ (√tan x + √cot x)dx, where x ∈ (0, π 2 ) ∪ (π, 3π 2 ) is : (A) √2tan−1 ( √tan x−√cot x √2 ) + C (B) √2tan−1 ( √tan x+√cot x √2 ) + C (C) − √2tan−1 ( √tan x−√cot x √2 ) + C (D) − √2tan−1 ( √tan x+√cot x √2 ) + C 47. Value of the integral I = ∫ (√tan x + √cot x)dx, where x ∈ (0, π 2 ) is : (A) √2sin−1 (cos x − sin x) + C (B) √2sin−1 (sin x − cos x) + C (C) √2sin−1 (sin x + cos x) + C (D) −√2sin−1 (sin x + cos x) + C 48. Value of the integral I = ∫ (√tan x + √cot x)dx, where x ∈ (π, 3π 2 ), is : (A) √2sin−1 (cos x − sin x) + C (B) √2sin−1 (sin x − cos x) + C (C) √2sin−1 (sin x + cos x) + C (D) − √2sin−1 (sin x + cos x) + C 49. For any natural number m, the value of ∫ (x 3m + x 2m + x m)(2x 2m + 3x m + 6) 1/mdx, x > 0 is : (A) 1 6(m+1) ⋅ (2x 3m + 3x 2m + 6x m) (m+1)/m + C