Integration – Introduction MCQs With Answer

by

Integration is a core part of calculus used extensively in B. Pharm studies, especially in pharmacokinetics and drug dosage calculations. This introduction covers basic integration concepts—indefinite and definite integrals, antiderivatives, integration rules (power rule, substitution, integration by parts), and applications like calculating the area under the plasma concentration–time curve (AUC). You will see examples of analytical integration and numerical techniques (trapezoidal rule, Simpson’s rule) used in drug concentration calculations. The MCQs focus on integration formulas, techniques, and pharmacy-related applications to prepare you for exams and practical tasks. Keywords include Integration, MCQs, B. Pharm, pharmacokinetics, AUC, numerical integration, antiderivative. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is an antiderivative of a function f(x)?

  • A function whose derivative is not related to f(x)
  • A function whose derivative equals f(x)
  • A function whose integral is zero
  • A function obtained by differentiating f(x)

Correct Answer: A function whose derivative equals f(x)

Q2. The Fundamental Theorem of Calculus connects differentiation and integration by stating:

  • The derivative of an integral with variable upper limit equals the integrand evaluated at that limit
  • The integral of a derivative is always zero
  • All integrals are antiderivatives multiplied by a constant
  • Definite integrals cannot be evaluated using antiderivatives

Correct Answer: The derivative of an integral with variable upper limit equals the integrand evaluated at that limit

Q3. Which statement distinguishes definite and indefinite integrals?

  • Definite integrals produce a family of functions; indefinite integrals produce a number
  • Indefinite integrals include a constant of integration; definite integrals evaluate to a number
  • Definite integrals always diverge; indefinite integrals always converge
  • Indefinite integrals require limits of integration

Correct Answer: Indefinite integrals include a constant of integration; definite integrals evaluate to a number

Q4. Using the power rule, what is ∫ x^4 dx ?

  • 4x^3 + C
  • x^5 + C
  • x^5 / 5 + C
  • ln|x| + C

Correct Answer: x^5 / 5 + C

Q5. What is ∫ e^x dx ?

  • e^x + C
  • e^x / x + C
  • ln(e^x) + C
  • x e^x + C

Correct Answer: e^x + C

Q6. What is the integral of 1/x with respect to x (for x ≠ 0)?

  • 1 / x + C
  • ln|x| + C
  • x + C
  • e^x + C

Correct Answer: ln|x| + C

Q7. What is ∫ sin x dx ?

  • −cos x + C
  • cos x + C
  • sin x + C
  • −sin x + C

Correct Answer: −cos x + C

Q8. Which substitution is appropriate to evaluate ∫ 2x cos(x^2) dx ?

  • u = cos x
  • u = x^2
  • u = 2x
  • u = sin x

Correct Answer: u = x^2

Q9. Which formula represents integration by parts?

  • ∫ u dv = uv − ∫ v du
  • ∫ u dv = u + v + C
  • ∫ u dv = ∫ du ∫ dv
  • ∫ u dv = uv + ∫ v du

Correct Answer: ∫ u dv = uv − ∫ v du

Q10. What is ∫ sec^2 x dx ?

  • tan x + C
  • sec x + C
  • ln|sec x| + C
  • cos x + C

Correct Answer: tan x + C

Q11. The trapezoidal rule is used for:

  • Symbolically integrating transcendental functions only
  • Numerically approximating definite integrals using trapezoids
  • Transforming integrals into differential equations
  • Calculating antiderivatives in closed form

Correct Answer: Numerically approximating definite integrals using trapezoids

Q12. Simpson’s rule requires how many subintervals to be applied correctly?

  • An odd number of subintervals
  • An even number of subintervals
  • Exactly one subinterval
  • Any prime number of subintervals

Correct Answer: An even number of subintervals

Q13. In pharmacokinetics, the area under the plasma concentration–time curve (AUC) is represented mathematically by:

  • The derivative of concentration over time
  • A definite integral of concentration with respect to time
  • The sum of concentration values only
  • The integral of dose with respect to volume

Correct Answer: A definite integral of concentration with respect to time

Q14. What are the typical units of AUC (area under concentration–time curve)?

  • mg·L / h
  • mg / (L·h)
  • concentration × time, e.g., mg·hr/L
  • time / concentration

Correct Answer: concentration × time, e.g., mg·hr/L

Q15. An improper integral is one where:

  • The integrand is a polynomial
  • The limits are finite and integrand continuous
  • The interval is infinite or integrand is unbounded on interval
  • It is always equal to zero

Correct Answer: The interval is infinite or integrand is unbounded on interval

Q16. Partial fraction decomposition is most useful to integrate:

  • Polynomial functions only
  • Trigonometric integrals only
  • Rational functions where denominator can be factored
  • Exponential functions like e^{ax}

Correct Answer: Rational functions where denominator can be factored

Q17. What is ∫ cos x dx ?

  • sin x + C
  • −sin x + C
  • cos x + C
  • tan x + C

Correct Answer: sin x + C

Q18. The antiderivative of a polynomial term 6x^2 is:

  • 2x^3 + C
  • 3x^3 + C
  • x^3 + C
  • 6x^3 + C

Correct Answer: 2x^3 + C

Q19. What is ∫ ln x dx ?

  • x ln x − x + C
  • ln x + C
  • x / ln x + C
  • 1 / x + C

Correct Answer: x ln x − x + C

Q20. Evaluate ∫ 2x e^{x^2} dx.

  • e^{x^2} + C
  • x e^{x^2} + C
  • 2 e^{x^2} + C
  • e^{2x} + C

Correct Answer: e^{x^2} + C

Q21. Evaluate ∫ x ln x dx (indefinite integral).

  • (x^2/2) ln x − x^2/4 + C
  • x ln x + C
  • (ln x)^2 / 2 + C
  • x^2 ln x + C

Correct Answer: (x^2/2) ln x − x^2/4 + C

Q22. What is the value of the definite integral ∫_0^1 x^2 dx ?

  • 1/2
  • 1/3
  • 1
  • 2/3

Correct Answer: 1/3

Q23. As the number of subintervals increases in numerical integration, the approximation generally:

  • Becomes less accurate
  • Remains unchanged
  • Becomes more accurate
  • Oscillates unpredictably

Correct Answer: Becomes more accurate

Q24. According to the fundamental theorem of calculus (part 2), ∫_a^b f'(x) dx equals:

  • f(b) − f(a)
  • f(a) + f(b)
  • f(b) × f(a)
  • 0 for all functions

Correct Answer: f(b) − f(a)

Q25. What is ∫ 1/(1 + x^2) dx ?

  • arctan x + C
  • ln(1 + x^2) + C
  • 1/(1 + x^2) + C
  • sin^{-1} x + C

Correct Answer: arctan x + C

Q26. What is ∫ e^{3x} dx ?

  • e^{3x} + C
  • (1/3) e^{3x} + C
  • 3 e^{3x} + C
  • ln(e^{3x}) + C

Correct Answer: (1/3) e^{3x} + C

Q27. For concentration-time data measured at discrete times, AUC is often estimated using:

  • Integration by parts]
  • Trapezoidal rule
  • Taylor series expansion
  • Laplace transforms

Correct Answer: Trapezoidal rule

Q28. Simpson’s rule typically gives more accurate results than the trapezoidal rule because it:

  • Approximates the integrand by a quadratic on subintervals
  • Uses smaller step sizes by default
  • Converts integrals to derivatives
  • Only uses the endpoints of intervals

Correct Answer: Approximates the integrand by a quadratic on subintervals

Q29. What is ∫ cos^2 x dx ?

  • (x/2) + (sin 2x)/4 + C
  • (x/2) − (sin 2x)/4 + C
  • sin x cos x + C
  • tan x + C

Correct Answer: (x/2) + (sin 2x)/4 + C

Q30. Evaluate ∫ 1/(x^2 − 1) dx (assuming |x|>1).

  • ln|x^2 − 1| + C
  • (1/2) ln|(x − 1)/(x + 1)| + C
  • arctan x + C
  • 1/(x − 1) + C

Correct Answer: (1/2) ln|(x − 1)/(x + 1)| + C

Q31. Indefinite integrals include an arbitrary constant because:

  • Every antiderivative differs by a constant
  • Integration is not well defined
  • Definite integrals require it
  • It makes the integral unique

Correct Answer: Every antiderivative differs by a constant

Q32. If F(x) = ∫_a^x f(t) dt, then F'(x) equals:

  • 0
  • f(x)
  • ∫_a^x f'(t) dt
  • F(a)

Correct Answer: f(x)

Q33. The area between two curves y = f(x) and y = g(x) from a to b is given by:

  • ∫_a^b f(x) g(x) dx
  • ∫_a^b |f(x) + g(x)| dx
  • ∫_a^b [f(x) − g(x)] dx when f(x) ≥ g(x)
  • ∫_a^b f'(x) − g'(x) dx

Correct Answer: ∫_a^b [f(x) − g(x)] dx when f(x) ≥ g(x)

Q34. A definite integral can be interpreted as:

  • The net signed area under a curve between limits
  • The slope of the tangent line at a point
  • The maximum value of the function
  • The limit of a sequence of derivatives

Correct Answer: The net signed area under a curve between limits

Q35. In pharmacokinetics, mean residence time (MRT) is calculated as:

  • AUC / AUMC
  • AUMC / AUC
  • Cmax / Tmax
  • Clearance × Volume

Correct Answer: AUMC / AUC

Q36. What is ∫ cos(2x) dx ?

  • (1/2) sin(2x) + C
  • sin(2x) + C
  • −(1/2) cos(2x) + C
  • 2 sin x + C

Correct Answer: (1/2) sin(2x) + C

Q37. The improper integral ∫_1^∞ 1/x^p dx converges if and only if:

  • p < 0
  • p ≤ 1
  • p > 1
  • All p are acceptable

Correct Answer: p > 1

Q38. When applying integration by parts to ∫ x e^x dx, a good choice is:

  • u = e^x, dv = x dx
  • u = x, dv = e^x dx
  • u = 1, dv = x e^x dx
  • u = xe^x, dv = dx

Correct Answer: u = x, dv = e^x dx

Q39. The global error of the trapezoidal rule decreases proportionally to which power of the step size h?

  • h
  • h^2
  • h^3
  • h^4

Correct Answer: h^2

Q40. Evaluate the definite integral ∫_0^{π} sin x dx .

  • 0
  • 1
  • 2
  • π

Correct Answer: 2

Q41. What is ∫ sec x tan x dx ?

  • sec x + C
  • tan x + C
  • ln|sec x| + C
  • cos x + C

Correct Answer: sec x + C

Q42. Which technique is most suitable to evaluate ∫ e^x sin x dx ?

  • Direct substitution only
  • Integration by parts twice
  • Partial fractions
  • Trapezoidal rule

Correct Answer: Integration by parts twice

Q43. What is ∫ x^{−1/2} dx ?

  • 2√x + C
  • √x + C
  • −2√x + C
  • ln|x| + C

Correct Answer: 2√x + C

Q44. What is the antiderivative of 3x^2 ?

  • x^3 + C
  • 3x^3 + C
  • x^2 + C
  • x^4/4 + C

Correct Answer: x^3 + C

Q45. Evaluate ∫_0^1 e^x dx .

  • 1
  • e
  • e − 1
  • ln e

Correct Answer: e − 1

Q46. The definite integral of an odd function over symmetric limits [−a, a] is:

  • Twice the integral from 0 to a
  • Zero
  • Equal to the integral from 0 to a
  • Undefined

Correct Answer: Zero

Q47. For a single IV bolus concentration C(t) = C0 e^{−k t}, AUC from 0 to ∞ equals:

  • C0 × k
  • C0 / k
  • k / C0
  • C0 × e^{−k}

Correct Answer: C0 / k

Q48. Which property of integrals states ∫_a^b [c f(x)] dx = c ∫_a^b f(x) dx ?

  • Linearity with respect to addition
  • Linearity with respect to scalar multiplication
  • Symmetry property
  • Monotonicity property

Correct Answer: Linearity with respect to scalar multiplication

Q49. The error term for Simpson’s rule is proportional to which power of step size h?

  • h^2
  • h^3
  • h^4
  • h^5

Correct Answer: h^4

Q50. Which expression correctly states the Fundamental Theorem of Calculus (part 2)?

  • If F is any antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) + F(a)
  • If F is differentiable and F’ = f, then ∫_a^b f(x) dx = F(b) − F(a)
  • ∫_a^b f(x) dx = f(b) − f(a)
  • The definite integral equals the derivative evaluated at the midpoint

Correct Answer: If F is differentiable and F’ = f, then ∫_a^b f(x) dx = F(b) − F(a)

Leave a Comment