Definite Integrals

Definite Integrals MCQs (1–39) with Solutions | Important Objective Questions PDF

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Practice 39 important Multiple Choice Questions on Definite Integrals with step-wise answers. Covers previous year papers (IIT, MP PET, AIEEE, KCET, BCECE, TGT PGT LT GRADE KVS NVSetc.). Free mobile-friendly quiz for students preparing JEE, Engineering & Competitive Exams.

Definite Integrals — MCQ (Q1–Q39)

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1) Evaluate ∫02/3 dx / (4 + 9x²)

  1. 0
  2. π/6
  3. π/12
  4. π/24
Show Answer
(d) π/24

2) Evaluate ∫0π/2 [sin x · cos x] / [cos²x + 3cos x + 2] dx

  1. log(3/4)
  2. log(4/7)
  3. log(8/7)
  4. log(9/8)
Show Answer
(d) log(9/8)

3) Evaluate ∫01 dx / (eˣ + e⁻ˣ)

  1. π/4
  2. tan⁻¹((e²+1)/(e−1))
  3. tan⁻¹((e−1)/(e+1))
  4. tan⁻¹((1−e)/(1+e))
Show Answer
(c) tan⁻¹((e−1)/(e+1))

4) Solve ∫3x √(t+1) dt = 0

  1. x=0
  2. x=1
  3. x=3
  4. x=5
Show Answer
(c) x = 3

5) Evaluate ∫01 √[(1−x)/(1+x)] dx

  1. 1/2
  2. π/2 − 1
  3. π²/4 + 1
  4. π²/2 − 1/3
Show Answer
(b) π/2 − 1

6) Evaluate ∫04 (x² − 2x + √x) dx

  1. 32/5
  2. 5/31
  3. 32/3
  4. 4/37
Show Answer
(c) 32/3

7) Evaluate ∫π/4π/2 cosec²x dx

  1. −1/2
  2. 0
  3. 1/2
  4. 1
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(d) 1

8) If ∫01 f(x)dx=1, ∫01 x f(x)dx=a, ∫01 x² f(x)dx=a², then ∫01 (a−x)² f(x)dx = ?

  1. 0
  2. a²
  3. 2a²
  4. 4a²
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(a) 0

9) Evaluate (determinant-integral as printed) from 0 to π/6

  1. 1/3
  2. 7/4
  3. 4/3
  4. π/2
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(c) 4/3

10) Evaluate ∫0π/2 dx / (a² cos²x + b² sin²x)

  1. πab
  2. π²ab
  3. π/(a²b²)
  4. π/(2ab)
Show Answer
(d) π/(2ab)

11) Evaluate ∫0π/2 dx / (4 cos²x + 9 sin²x)

  1. π/18
  2. π/12
  3. π/6
  4. π/2
Show Answer
(b) π/12

12) Evaluate ∫0π/2 (sin x + cos x)² / √(1 + sin 2x) dx

  1. 3
  2. 2
  3. 1
  4. 0
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(b) 2

13) Evaluate ∫0π [sin x + cos x] / √(1 + sin 2x) · (with π factor as printed) dx

  1. π
  2. π/2
  3. π/4
  4. π/6
Show Answer
(a) π

14) Evaluate ∫01/2 [x·sin⁻¹x]/√(1−x²) dx

  1. 1/2 + (π√3)/12
  2. 1/2 + π/(12√3)
  3. 1/2 − π/(3√3)
  4. 1/2 − π/(4√3)
Show Answer
(d)

15) Evaluate ∫01/√2 [sin⁻¹x]/[(1−x²)√(1−x²)] dx

  1. π/4 − √2·log2
  2. π/4·log2 + 2
  3. π/4 + (1/2)log2
  4. π/4 − log√2
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(d)

16) Evaluate ∫0π/2 x·sin x dx

  1. π/4
  2. π/2
  3. π
  4. 1
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(d) 1

17) If f(y)=eʸ, g(y)=y, y>0 and F(t)=∫0t f(t−y)g(y) dy, then F(t)= ?

  1. −t−1
  2. t·e⁻¹
  3. eᵗ − (1+t)
  4. t·eᵗ + (1−t)
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(c)

18) If Iₙ=∫0π/2 [sin(2n+1)x]/[sin x] dx, find Iₙ − Iₙ₋₁

  1. 0
  2. 1
  3. 2
  4. π/2
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(a) 0

19) Evaluate ∫01 (x⁴+1)/(x²+1) dx

  1. (1/6)(3π−4)
  2. (1/6)(3−4π)
  3. (1/6)(3π+4)
  4. (1/6)(3+4π)
Show Answer
(a)

20) Evaluate ∫0π/4 x·sec²x dx

  1. π/4 − 8
  2. π/4 − 3 log 2
  3. π/4 − (1/2) log 2
  4. π/4 − (1/3) log 2
Show Answer
(c)

21) Evaluate ∫0π/4 tan²x dx

  1. π/2 − 1
  2. 1 − π/4
  3. 1 + π/4
  4. π/8 − 1
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(b)

22) Evaluate ∫12 log x dx

  1. log(e/2)
  2. log(2/e)
  3. log(e²/4)
  4. log(4/e)
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(d)

23) Evaluate ∫35 x²/(x²−4) dx

  1. 2 − log(14/15)
  2. 2 + log(15/7)
  3. 2 − tan⁻¹(15/7)
  4. …
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(b)

24) If ∫0α dx/(1+4x²) = π/8, then α = ?

  1. 1/2
  2. π/2
  3. 1
  4. π
Show Answer
(a) 1/2

25) Evaluate ∫0π/2 sin x · sin 2x dx

  1. 4/3
  2. 1/3
  3. 3/4
  4. 2/3
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(d)

26) Evaluate ∫ab (log x)/x dx

  1. ½[(log b + log a)/(log b − log a)]
  2. ½ log(b/a) · log(ab)
  3. ½ log((b²−a²)/(b²+a²))
  4. ½[log(b+a)/log(b−a)]
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(b)

27) Evaluate ∫1e (1/x) dx

  1. 0
  2. 1
  3. e
  4. log(e−1)
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(b) 1

28) Evaluate ∫01 dx / [(1+x²)√(2+x²)]

  1. π/2
  2. π/3
  3. π/6
  4. π/9
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(c) π/6

29) Evaluate ∫0π/2 e^{sin x} · cos x dx

  1. e
  2. e−1
  3. ½(e−1)
  4. ½(e−2)
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(b)

30) Evaluate ∫12 eˣ(1/x − 1/x²) dx

  1. 0
  2. (e/2)(e−1)
  3. e(e−1)
  4. e(e/2 − 1)
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(d)

31) Evaluate ∫12 [eˣ(1 + x·log x)]/x dx

  1. e² log 2
  2. ½(e² − 2e)
  3. ½(e² + 2e)
  4. ½(e² + log 2)
Show Answer
(a)

32) Evaluate ∫01 [x eˣ]/(1+x)² dx

  1. 2e − 1
  2. e − 2
  3. ½(e − 2)
  4. ½(e + 2)
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(c)

33) Evaluate ∫0π/2 eˣ · (1 + sin x)/(1 + cos x) dx

  1. 0
  2. π/4
  3. e^{π/2}
  4. e^{π/2} − 1
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(c)

34) Evaluate ∫−11/2 [eˣ(2 − x²)]/[(1−x)√(1−x²)] dx

  1. √3 · e
  2. e/√3
  3. √(3e)/2
  4. √e/2 (√3 + 1)
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(a)

35) Evaluate ∫π/2π eˣ · (1 − sin x)/(1 − cos x) dx

  1. e^π
  2. e^{π/2}
  3. e^{π/4}
  4. e^{−π/2}
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(b)

36) Value of ∫−22 (ax³ + bx + c) dx depends on?

  1. a
  2. b
  3. c
  4. a and b
Show Answer
(c) only c

37) Evaluate ∫0π/4 (sinθ + cosθ)/(3 + sin2θ) dθ

  1. ½ log 8
  2. ¼ log 3
  3. ¼ log(1/2)
  4. 1/3 log(1/4)
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(b)

38) Evaluate ∫0π/4 (√tan x + √cot x) dx

  1. 2π/√3
  2. π/√2
  3. π/(2√2)
  4. π/(2√3)
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(b)

39) Evaluate ∫0π/4 (cos x + sin x)/(9 + 16 sin 2x) dx

  1. (1/15) log 5
  2. (1/5) log 10
  3. (1/20) log 3
  4. (1/10) log 4
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(c)
Answer Key: 1(d), 2(d), 3(c), 4(c), 5(b), 6(c), 7(d), 8(a), 9(c), 10(d), 11(b), 12(b), 13(a), 14(d), 15(d), 16(d), 17(c), 18(a), 19(a), 20(c), 21(b), 22(d), 23(b), 24(a), 25(d), 26(b), 27(b), 28(c), 29(b), 30(d), 31(a), 32(c), 33(c), 34(a), 35(b), 36(c), 37(b), 38(b), 39(c).

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