Definition of integration MCQs With Answer

Definition of integration MCQs With Answer provides B. Pharm students a clear, focused review of integral calculus concepts and their pharmaceutical applications. This Student-friendly post covers key terms like definite integral, indefinite integral, fundamental theorem of calculus, substitution, integration by parts, numerical integration, and AUC in pharmacokinetics. Designed for B. Pharm students, the content links mathematical definitions to drug dosage calculations, area under the curve (AUC), and rate processes in pharmaceutics. These integration MCQs with answers boost understanding, exam readiness, and practical problem-solving skills. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the definition of an indefinite integral?

• A numerical value representing the area under a curve between two points
• A family of functions whose derivative gives the integrand
• A method for approximating definite integrals using rectangles
• A special type of differential equation solution method

Correct Answer: A family of functions whose derivative gives the integrand

Q2. Which statement best defines a definite integral?

• An antiderivative of a function without limits
• The signed area under a curve between two specified limits
• A sequence of approximations converging to a derivative
• A formula for integrating rational functions only

Correct Answer: The signed area under a curve between two specified limits

Q3. What does the Fundamental Theorem of Calculus link?

• Limits and infinite series
• Differentiation and integration
• Partial fractions and trigonometric substitution
• Numerical methods and error estimation

Correct Answer: Differentiation and integration

Q4. In pharmacokinetics, what does AUC represent?

• Absolute uptake coefficient of a drug
• Area under the plasma concentration-time curve
• Average urinary clearance
• Approximate unbound concentration

Correct Answer: Area under the plasma concentration-time curve

Q5. Which integration technique is most appropriate for ∫ x e^{x} dx?

• Partial fractions
• Integration by parts
• Trigonometric substitution
• Direct antiderivative lookup only

Correct Answer: Integration by parts

Q6. How is the area under a concentration-time curve from 0 to ∞ commonly estimated?

• Using repeated differentiation
• Numerical integration plus extrapolation from last measured point
• Solving a linear algebraic equation
• Applying the Laplace transform only

Correct Answer: Numerical integration plus extrapolation from last measured point

Q7. Which substitution is useful for ∫ (2x)/(x^2+1) dx?

• u = x^2 + 1
• u = 2x
• u = 1/x
• u = arctan x

Correct Answer: u = x^2 + 1

Q8. What is an improper integral?

• An integral with a discontinuous integrand or infinite limits
• An integral that cannot be solved by substitution
• An integral only solvable by numerical methods
• An integral representing steady-state concentration

Correct Answer: An integral with a discontinuous integrand or infinite limits

Q9. Which numerical method uses parabolic arcs to approximate integrals?

• Trapezoidal rule
• Simpson’s rule
• Left Riemann sum
• Monte Carlo integration

Correct Answer: Simpson’s rule

Q10. For a first-order elimination process C(t)=C0 e^{-kt}, what is the integral of C(t) from 0 to ∞?

• C0/k
• C0*k
• k/C0
• Zero

Correct Answer: C0/k

Q11. Which rule gives the derivative of an integral with a variable upper limit?

• Integration by parts
• Fundamental theorem of calculus
• Partial fraction decomposition
• Mean value theorem for integrals

Correct Answer: Fundamental theorem of calculus

Q12. When integrating rational functions, what technique often simplifies the integrand?

• Integration by parts
• Partial fraction decomposition
• Trigonometric substitution for linear terms
• Using the Laplace transform directly

Correct Answer: Partial fraction decomposition

Q13. The indefinite integral ∫ cos x dx equals:

• sin x + C
• -sin x + C
• cos x + C
• -cos x + C

Correct Answer: sin x + C

Q14. Which integral property helps split integrals over adjacent intervals?

• Linearity of integration and additivity over intervals
• Integration by substitution
• Integration by parts
• Change of variables only for improper integrals

Correct Answer: Linearity of integration and additivity over intervals

Q15. For definite integrals, linearity refers to:

• ∫(a f + b g) = a ∫f + b ∫g
• Changing limits flips sign only
• Using only constant multiples is allowed
• Integration and differentiation commute always

Correct Answer: ∫(a f + b g) = a ∫f + b ∫g

Q16. Which substitution is best for ∫ sin^3 x cos x dx?

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

Correct Answer: u = sin x

Q17. Integration by parts is derived from which product rule?

• Differentiation of a quotient
• Differentiation of a product
• Chain rule
• Leibniz rule for higher derivatives

Correct Answer: Differentiation of a product

Q18. How is clearance (Cl) related to AUC for an IV bolus?

• Cl = Dose × AUC
• Cl = Dose / AUC
• Cl = AUC / Dose
• Cl = Dose + AUC

Correct Answer: Cl = Dose / AUC

Q19. Which integral represents the convolution of input function f(t) with system response g(t)?

• ∫ f(t) g(t) dt from 0 to ∞
• ∫ f(τ) g(t-τ) dτ from 0 to t
• ∫ f(t) dt × ∫ g(t) dt
• ∫ f(t)/g(t) dt

Correct Answer: ∫ f(τ) g(t-τ) dτ from 0 to t

Q20. Which method is preferred to evaluate ∫ dx/(x^2-1)?

• Trigonometric substitution
• Partial fraction decomposition
• Integration by parts
• Simpson’s rule

Correct Answer: Partial fraction decomposition

Q21. What is the integral of 1/x dx (x>0)?

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

Correct Answer: ln |x| + C

Q22. In drug release kinetics, which integral helps compute cumulative drug released over time?

• Definite integral of release rate from 0 to t
• Indefinite integral without constant
• Derivative of plasma concentration
• Inverse Laplace transform only

Correct Answer: Definite integral of release rate from 0 to t

Q23. For definite integrals, reversing the limits changes the sign. True or false?

• True
• False
• Only true for symmetric limits
• Only true for improper integrals

Correct Answer: True

Q24. Which integrand is best handled by trigonometric substitution?

• 1/(x+1)
• 1/√(a^2 – x^2)
• e^{x^2}
• ln x

Correct Answer: 1/√(a^2 – x^2)

Q25. What is the integral ∫ 0 to T k e^{-kt} dt equal to?

• e^{-kT}
• 1 – e^{-kT}
• kT e^{-kT}
• T/k

Correct Answer: 1 – e^{-kT}

Q26. When using u-substitution, dx is replaced by:

• du / (du/dx)
• du × (du/dx)
• du / (dx/du)
• du

Correct Answer: du / (du/dx)

Q27. Which integrand requires using the substitution u = tan x? (Hint: involves sec^2 x)

• ∫ sec^2 x dx
• ∫ sin x dx
• ∫ cos x dx
• ∫ 1/x dx

Correct Answer: ∫ sec^2 x dx

Q28. The mean residence time (MRT) involves which integral expression?

• MRT = ∫ t C(t) dt / ∫ C(t) dt
• MRT = ∫ C(t) dt × ∫ t dt
• MRT = ∫ C(t)/t dt
• MRT = ∫ dC/dt dt

Correct Answer: MRT = ∫ t C(t) dt / ∫ C(t) dt

Q29. For integrating ∫ e^{ax} dx, the antiderivative is:

• (1/a) e^{ax} + C
• a e^{ax} + C
• e^{ax} + C
• ln(e^{ax}) + C

Correct Answer: (1/a) e^{ax} + C

Q30. Which statement about numerical integration errors is correct?

• Simpson’s rule often gives higher accuracy than trapezoidal for smooth functions
• Trapezoidal rule is always more accurate than Simpson’s rule
• Numerical methods produce no error with irregular functions
• Errors cannot be estimated

Correct Answer: Simpson’s rule often gives higher accuracy than trapezoidal for smooth functions

Q31. Which integral is evaluated using the substitution x = a sec θ?

• ∫ dx/√(x^2 – a^2)
• ∫ dx/√(a^2 – x^2)
• ∫ dx/(x^2+1)
• ∫ e^{x^2} dx

Correct Answer: ∫ dx/√(x^2 – a^2)

Q32. What is the antiderivative of 2x/(1+x^2)?

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

Correct Answer: ln(1+x^2) + C

Q33. When is it appropriate to use integration by parts twice?

• For integrands like x^2 e^{x} or x^2 cos x
• When integrand is rational with linear denominator
• Only for definite integrals
• Never; use substitution instead

Correct Answer: For integrands like x^2 e^{x} or x^2 cos x

Q34. To compute AUC from discrete concentration-time points, which technique is commonly used?

• Analytical integration only
• Linear trapezoidal rule
• Integration by parts
• Partial fractions

Correct Answer: Linear trapezoidal rule

Q35. Which integral represents the total amount eliminated from time 0 to t for elimination rate k and concentration C(t)?

• ∫ 0 to t k C(t) dt
• ∫ 0 to t C(t)/k dt
• C(t)/k
• k ∫ 0 to t dt

Correct Answer: ∫ 0 to t k C(t) dt

Q36. Which antiderivative corresponds to ∫ sec^2 x dx?

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

Correct Answer: tan x + C

Q37. For ∫ (ax+b) / (cx+d) dx, what method simplifies the integral?

• Long division followed by simple substitution
• Trigonometric substitution
• Integration by parts repeatedly
• Simpson’s rule

Correct Answer: Long division followed by simple substitution

Q38. Which integral test determines convergence of improper integrals?

• Comparison test with known convergent integrals
• Integration by parts always
• Trapezoidal error test
• Simpson convergence theorem

Correct Answer: Comparison test with known convergent integrals

Q39. Which integral equals zero by symmetry when integrand is odd and limits are symmetric?

• ∫_{-a}^{a} f(x) dx if f is even
• ∫_{-a}^{a} f(x) dx if f is odd
• ∫_{0}^{a} f(x) dx if f is odd
• None of the above

Correct Answer: ∫_{-a}^{a} f(x) dx if f is odd

Q40. When evaluating ∫ dx/(x^2+1), the antiderivative is:

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

Correct Answer: arctan x + C

Q41. In pharmacokinetics, the clearance concept can be derived from which integral relation?

• Area under the curve relating dose and systemic exposure
• Integral of volume over time
• Derivative of concentration at t=0
• Integral of molecular weight

Correct Answer: Area under the curve relating dose and systemic exposure

Q42. Which substitution simplifies ∫ dx/(√(x^2+a^2))?

• x = a sin θ
• x = a tan θ
• x = a cos θ
• x = a sec θ

Correct Answer: x = a tan θ

Q43. Which statement best describes the linearity of definite integrals?

• ∫ f+g = ∫ f × ∫ g
• ∫ (c f) = c ∫ f
• ∫ f^2 = (∫ f)^2
• Linearity applies only to improper integrals

Correct Answer: ∫ (c f) = c ∫ f

Q44. The area under a curve computed by Riemann sums becomes exact as:

• The number of partitions approaches zero
• The number of partitions approaches infinity and mesh size tends to zero
• The function becomes discontinuous
• The integrand increases without bound

Correct Answer: The number of partitions approaches infinity and mesh size tends to zero

Q45. Which integral is easiest evaluated by recognizing a derivative of a logarithm?

• ∫ (2x)/(1+x^2) dx
• ∫ x^2 dx
• ∫ e^{x} dx
• ∫ sin x dx

Correct Answer: ∫ (2x)/(1+x^2) dx

Q46. In calculating AUC using the trapezoidal rule, what assumption is made between measured points?

• Concentration varies exponentially
• Concentration varies linearly between points
• Concentration stays constant
• Concentration follows a parabolic path always

Correct Answer: Concentration varies linearly between points

Q47. What is the antiderivative of cosh x?

• sinh x + C
• cosh x + C
• e^{x} + C
• tanh x + C

Correct Answer: sinh x + C

Q48. For integrals containing √(a^2 – x^2), which trig substitution is used?

• x = a sec θ
• x = a tan θ
• x = a sin θ
• x = a cosh u

Correct Answer: x = a sin θ

Q49. Which integral technique helps evaluate ∫ e^{x} sin x dx?

• Single substitution
• Integration by parts twice
• Partial fractions
• Trigonometric substitution

Correct Answer: Integration by parts twice

Q50. How does integration apply to calculating cumulative bioavailability from rate data?

• By differentiating the rate to get concentration
• By integrating the absorption rate over time to get total absorbed
• By taking the average of rate values only
• By applying partial fractions to the rate directly

Correct Answer: By integrating the absorption rate over time to get total absorbed

G S Sachin

I am a Registered Pharmacist under the Pharmacy Act, 1948, and the founder of PharmacyFreak.com. I hold a Bachelor of Pharmacy degree from Rungta College of Pharmaceutical Science and Research. With a strong academic foundation and practical knowledge, I am committed to providing accurate, easy-to-understand content to support pharmacy students and professionals. My aim is to make complex pharmaceutical concepts accessible and useful for real-world application.

Mail- Sachin@pharmacyfreak.com