Definite integrals MCQs With Answer

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Introduction: Definite integrals MCQs With Answer provide B. Pharm students a focused way to master integral concepts applied in pharmacokinetics and pharmaceutical calculations. This concise, keyword-rich guide covers definite integrals, area under the curve (AUC), numerical integration (trapezoidal and Simpson), properties of definite integrals, improper integrals, and applications to drug concentration–time profiles. By practicing targeted multiple-choice questions, B. Pharm learners strengthen skills in evaluating integrals, interpreting AUC, estimating bioavailability, and using integral techniques for drug release and clearance calculations. Clear explanations and repetitive practice improve exam readiness and practical problem solving. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. Which property of definite integrals allows splitting an integral from a to c into two integrals a to b and b to c?

  • Linearity of integrals
  • Additivity over intervals
  • Fundamental theorem of calculus
  • Integration by parts

Correct Answer: Additivity over intervals

Q2. The definite integral ∫_0^T C(t) dt in pharmacokinetics represents which quantity?

  • Elimination rate constant
  • Area under the concentration-time curve (AUC)
  • Bioavailability fraction
  • Half-life

Correct Answer: Area under the concentration-time curve (AUC)

Q3. If f(x) is continuous, which theorem connects definite integrals and antiderivatives?

  • Mean value theorem for derivatives
  • Fundamental theorem of calculus
  • Rolle’s theorem
  • Green’s theorem

Correct Answer: Fundamental theorem of calculus

Q4. For an even function f(x), what simplification holds for ∫_{-a}^{a} f(x) dx?

  • 0
  • 2 ∫_{0}^{a} f(x) dx
  • ∫_{0}^{a} f(x) dx
  • -2 ∫_{0}^{a} f(x) dx

Correct Answer: 2 ∫_{0}^{a} f(x) dx

Q5. For an odd function f(x), the value of ∫_{-a}^{a} f(x) dx is:

  • Equal to ∫_{0}^{a} f(x) dx
  • Twice the integral from 0 to a
  • Zero
  • Undefined

Correct Answer: Zero

Q6. Which numerical method is most accurate for smooth functions using an even number of subintervals?

  • Left Riemann sum
  • Trapezoidal rule
  • Simpson’s rule
  • Right Riemann sum

Correct Answer: Simpson’s rule

Q7. The trapezoidal rule approximates integrals by summing areas of which shape?

  • Rectangles
  • Triangles
  • Trapezoids
  • Circles

Correct Answer: Trapezoids

Q8. For concentration C(t)=C0 e^{-kt}, the AUC from 0 to ∞ equals:

  • C0 / k
  • k / C0
  • C0 * k
  • 0

Correct Answer: C0 / k

Q9. In one-compartment IV bolus model, C(t)=C0 e^{-kt}. The definite integral ∫_{0}^{T} C(t) dt gives:

  • Total drug eliminated in [0,T]
  • Average dose
  • Drug clearance
  • Area under the curve from 0 to T

Correct Answer: Area under the curve from 0 to T

Q10. The improper integral ∫_{0}^{∞} e^{-αt} dt converges if:

  • α > 0
  • α = 0
  • α < 0
  • Always converges

Correct Answer: α > 0

Q11. Which substitution is useful to evaluate ∫_{0}^{1} x e^{x^2} dx?

  • u = x
  • u = x^2
  • u = e^{x^2}
  • u = 1/x

Correct Answer: u = x^2

Q12. The mean value theorem for definite integrals guarantees existence of c in [a,b] such that ∫_{a}^{b} f(x) dx = f(c)(b-a) when f is:

  • Discontinuous
  • Integrable only
  • Continuous
  • Differentiable only at endpoints

Correct Answer: Continuous

Q13. If F(x) is an antiderivative of f(x), then d/dx ∫_{a}^{x} f(t) dt equals:

  • F(x) – F(a)
  • f(x)
  • ∫_{a}^{x} F(t) dt
  • 0

Correct Answer: f(x)

Q14. When evaluating ∫_{0}^{π} sin x dx using symmetry, the result is:

  • 0
  • 2
  • 1
  • π

Correct Answer: 2

Q15. For pharmacokinetic data approximated at discrete times, which integral approach estimates AUC most directly?

  • Molecular orbital theory
  • Numerical integration such as trapezoidal rule
  • Solving differential equations analytically only
  • Using derivative values only

Correct Answer: Numerical integration such as trapezoidal rule

Q16. Integration by parts is derived from which rule?

  • Product rule for differentiation
  • Quotient rule
  • Chain rule
  • Power rule

Correct Answer: Product rule for differentiation

Q17. The definite integral ∫_{a}^{b} k dx where k is constant equals:

  • k
  • k(b-a)
  • (b-a)/k
  • 0

Correct Answer: k(b-a)

Q18. For evaluating ∫_{0}^{1} (1/(1+x^2)) dx, the antiderivative is:

  • arctan x
  • ln(1+x^2)
  • 1/(1+x^2)
  • tan x

Correct Answer: arctan x

Q19. Which definite integral expresses average concentration C_avg over interval [0,T]?

  • ∫_{0}^{T} C(t) dt
  • (1/T) ∫_{0}^{T} C(t) dt
  • C(T) – C(0)
  • ∫_{0}^{T} t C(t) dt

Correct Answer: (1/T) ∫_{0}^{T} C(t) dt

Q20. If ∫_{0}^{2} f(x) dx = 5 and ∫_{2}^{5} f(x) dx = 7, then ∫_{0}^{5} f(x) dx equals:

  • 2
  • 35
  • 12
  • Undefined

Correct Answer: 12

Q21. Which of the following integrals is improper?

  • ∫_{0}^{1} x^2 dx
  • ∫_{1}^{∞} 1/x^2 dx
  • ∫_{0}^{π} sin x dx
  • ∫_{-1}^{1} x dx

Correct Answer: ∫_{1}^{∞} 1/x^2 dx

Q22. The integral ∫_{0}^{T} ke^{-kt} dt with k>0 evaluates to:

  • 1 – e^{-kT}
  • e^{-kT}
  • kT
  • 0

Correct Answer: 1 – e^{-kT}

Q23. For AUC extrapolated to infinity from last time t_last with concentration C_last and ke, the added area equals:

  • C_last * ke
  • C_last / ke
  • ke / C_last
  • 0

Correct Answer: C_last / ke

Q24. To compute ∫ x cos x dx (definite), a useful technique is:

  • Substitution u = cos x
  • Integration by parts
  • Partial fractions
  • Trapezoidal rule only

Correct Answer: Integration by parts

Q25. The definite integral ∫_{a}^{b} f(x) dx changes sign if:

  • a and b are swapped
  • f(x) is scaled by 2
  • The function is shifted vertically
  • The integrand is multiplied by -1/2

Correct Answer: a and b are swapped

Q26. In clearance calculations, total drug exposure AUC is inversely proportional to which parameter for IV bolus?

  • Volume of distribution Vd
  • Clearance CL
  • Initial concentration C0
  • Absorption rate ka

Correct Answer: Clearance CL

Q27. Evaluate ∫_{0}^{1} 3x^2 dx.

  • 1
  • 3
  • 0
  • 1/3

Correct Answer: 1

Q28. Which substitution helps evaluate ∫_{0}^{π/2} sin^2 x dx using symmetry or identity?

  • Use identity sin^2 x = (1 – cos 2x)/2
  • u = cos x
  • u = tan x
  • u = x^2

Correct Answer: Use identity sin^2 x = (1 – cos 2x)/2

Q29. Simpson’s rule requires the number of subintervals to be:

  • Odd
  • Even
  • Prime
  • Zero

Correct Answer: Even

Q30. The error of the trapezoidal rule is proportional to the second derivative of f. This means better accuracy when f” is:

  • Large and oscillatory
  • Zero or small
  • Undefined
  • Highly discontinuous

Correct Answer: Zero or small

Q31. For definite integral ∫_{0}^{2π} cos x dx the value is:

  • 2π
  • 0
  • 1
  • -2π

Correct Answer: 0

Q32. The definite integral ∫_{a}^{a} f(x) dx equals:

  • f(a)
  • 0
  • Undefined
  • Infinity

Correct Answer: 0

Q33. If concentration C(t) is measured at t=0,1,2 hr as 10, 6, 3 µg/mL, trapezoidal AUC from 0 to 2 hr equals:

  • (1/2)*(10+6) + (1/2)*(6+3)
  • 10+6+3
  • (2)*(10+3)
  • (10-3)/2

Correct Answer: (1/2)*(10+6) + (1/2)*(6+3)

Q34. To evaluate ∫_{0}^{1} ln(1+x) dx, which integration method is appropriate?

  • Integration by parts
  • Direct substitution u=1+x only
  • Simpson’s rule only
  • Partial fractions

Correct Answer: Integration by parts

Q35. The definite integral ∫_{0}^{∞} t e^{-t} dt equals:

  • 0
  • 1
  • ∞
  • 2

Correct Answer: 1

Q36. For a concentration function C(t)=At/(B+t), evaluating ∫_{0}^{T} C(t) dt may require which technique?

  • Polynomial division and substitution
  • Integration by parts only
  • Fourier transform
  • Matrix inversion

Correct Answer: Polynomial division and substitution

Q37. The integral ∫_{0}^{1} (1/√x) dx is improper and equals:

  • 2
  • 1
  • 0
  • Divergent

Correct Answer: 2

Q38. Using substitution u = 1 + t^2 helps evaluate which integral?

  • ∫ t/(1+t^2) dt
  • ∫ e^{t} dt
  • ∫ sin t dt
  • ∫ 1/t dt

Correct Answer: ∫ t/(1+t^2) dt

Q39. The definite integral is linear: ∫_{a}^{b} [αf(x)+βg(x)] dx equals:

  • α ∫_{a}^{b} f(x) dx + β ∫_{a}^{b} g(x) dx
  • αβ ∫_{a}^{b} f(x)g(x) dx
  • ∫_{a}^{b} f(x) dx + ∫_{a}^{b} g(x) dx only if α=β=1
  • None of the above

Correct Answer: α ∫_{a}^{b} f(x) dx + β ∫_{a}^{b} g(x) dx

Q40. For a one-compartment oral model with first-order absorption ka, the AUC involves integrating which product?

  • Absorption constant times time only
  • Concentration-time profile C(t)
  • Volume times clearance
  • Rate constants difference only

Correct Answer: Concentration-time profile C(t)

Q41. When integrating definite integrals numerically, decreasing step size h generally:

  • Increases error
  • Decreases accuracy
  • Improves accuracy
  • Has no effect

Correct Answer: Improves accuracy

Q42. The definite integral ∫_{0}^{π/4} sec^2 x dx equals:

  • tan(π/4) – tan(0)
  • ln(sec x) evaluated
  • sin x evaluated
  • Undefined

Correct Answer: tan(π/4) – tan(0)

Q43. Which of the following expresses AUC from 0 to ∞ for first-order elimination with dose D and clearance CL (IV bolus)?

  • CL / D
  • D / CL
  • D * CL
  • 0

Correct Answer: D / CL

Q44. To evaluate ∫_{0}^{2} (2x+1) dx, the result is:

  • 8
  • 6
  • 5
  • 4

Correct Answer: 6

Q45. If f(x) ≥ 0 on [a,b], then ∫_{a}^{b} f(x) dx is:

  • Negative
  • Non-negative
  • Undefined
  • Always zero

Correct Answer: Non-negative

Q46. In drug release studies, the cumulative released amount Q(t) can be obtained by integrating which function?

  • Release rate r(t)
  • Initial dose only
  • Volume of solvent
  • Rate constant squared

Correct Answer: Release rate r(t)

Q47. The substitution x = a + (b-a)u transforms ∫_{a}^{b} f(x) dx into:

  • (b-a) ∫_{0}^{1} f(a+(b-a)u) du
  • ∫_{0}^{1} f(u) du only
  • ∫_{a}^{b} f(u) du
  • 0

Correct Answer: (b-a) ∫_{0}^{1} f(a+(b-a)u) du

Q48. The integral ∫_{0}^{1} x^n dx for integer n ≥ 0 equals:

  • 1/(n+1)
  • n/(n+1)
  • 0
  • n!

Correct Answer: 1/(n+1)

Q49. Which integral test relates improper integrals to series convergence?

  • Comparison test only
  • Integral test for series convergence
  • Rolle’s theorem
  • Mean value theorem

Correct Answer: Integral test for series convergence

Q50. For a given concentration function C(t), the harmonic mean concentration over [0,T] involves integrating which expression?

  • (1/T) ∫_{0}^{T} C(t) dt
  • T / ∫_{0}^{T} (1/C(t)) dt
  • ∫_{0}^{T} C(t)^2 dt
  • ∫_{0}^{T} ln C(t) dt

Correct Answer: T / ∫_{0}^{T} (1/C(t)) dt

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