Integration by parts MCQs With Answer

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Integration by parts MCQs With Answer is an essential resource for B.Pharm students learning advanced integration techniques and their applications in pharmacokinetics and pharmaceutics. This collection focuses on integration by parts theory, selection strategies (LIATE), reduction formulas, definite integrals, repeated and tabular methods, and real-world problems like AUC, mean residence times, and drug release profiles. Each question reinforces both technique and interpretation so you can solve symbolic integrals and apply results to dosage calculations, kinetics, and formulation modeling. Clear explanations and focused practice build the confidence needed for exams and research tasks. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the core formula for integration by parts?

  • ∫ u dv = uv + ∫ v du
  • ∫ u dv = uv – ∫ v du
  • ∫ u dv = u’v – ∫ u v’
  • ∫ u dv = ∫ v du – uv

Correct Answer: ∫ u dv = uv – ∫ v du

Q2. According to the LIATE rule for choosing u in integration by parts, which function type has highest priority?

  • Trigonometric
  • Algebraic (polynomial)
  • Inverse trigonometric
  • Logarithmic

Correct Answer: Logarithmic

Q3. For ∫ x e^{2x} dx, which choice of u simplifies the integral best?

  • u = e^{2x}, dv = x dx
  • u = x, dv = e^{2x} dx
  • u = 2x, dv = e^{x} dx
  • u = x e^{2x}, dv = dx

Correct Answer: u = x, dv = e^{2x} dx

Q4. Evaluate ∫ ln x dx. Which is the correct result?

  • ln x + C
  • x ln x – x + C
  • x / ln x + C
  • ln(x^2)/2 + C

Correct Answer: x ln x – x + C

Q5. Using integration by parts, the integral ∫ x^2 e^x dx requires how many iterations (naive count)?

  • One iteration
  • Two iterations
  • Three iterations
  • No iterations needed

Correct Answer: Two iterations

Q6. For definite integrals, the integration by parts formula includes boundary terms. Which expression is correct?

  • ∫_a^b u dv = [uv]_a^b + ∫_a^b v du
  • ∫_a^b u dv = [uv]_a^b – ∫_a^b v du
  • ∫_a^b u dv = ∫_a^b v du – [uv]_a^b
  • ∫_a^b u dv = -[uv]_a^b + ∫_a^b v du

Correct Answer: ∫_a^b u dv = [uv]_a^b – ∫_a^b v du

Q7. Which integral is best solved using integration by parts: ∫ x ln(x^2) dx?

  • No, use substitution only
  • Yes, set u = ln(x^2) and dv = x dx
  • Yes, set u = x and dv = ln(x^2) dx
  • No, it diverges

Correct Answer: Yes, set u = ln(x^2) and dv = x dx

Q8. The tabular method of integration by parts is most helpful when:

  • Both functions are trigonometric
  • One factor differentiates to zero after finite steps
  • Both functions are exponentials
  • The integrand is rational only

Correct Answer: One factor differentiates to zero after finite steps

Q9. ∫_0^1 x ln x dx equals?

  • -1/4
  • -1/2
  • -1/9
  • -1/4 + ln 1

Correct Answer: -1/4

Q10. Integration by parts applied to ∫ e^x sin x dx leads to a system that requires:

  • Only one application of parts
  • Two applications and solving algebraically
  • Infinite series expansion
  • Direct substitution

Correct Answer: Two applications and solving algebraically

Q11. Choose the best u for ∫ x cos^{-1}(x) dx (inverse cosine):

  • u = x, dv = cos^{-1}(x) dx
  • u = cos^{-1}(x), dv = x dx
  • u = 1, dv = x cos^{-1}(x) dx
  • u = cos(x), dv = x dx

Correct Answer: u = cos^{-1}(x), dv = x dx

Q12. Which of these integrals is NOT typically solved by integration by parts?

  • ∫ x e^{x} dx
  • ∫ ln x dx
  • ∫ e^{x^2} x dx
  • ∫ x^2 sin x dx

Correct Answer: ∫ e^{x^2} x dx

Q13. In pharmacokinetics, area under the curve (AUC) from t0 to t1 can be computed by integrating concentration-time data. If concentration C(t) = t e^{-t}, which method helps integrate ∫ t e^{-t} dt?

  • Integration by substitution only
  • Integration by parts
  • Partial fractions
  • Trigonometric substitution

Correct Answer: Integration by parts

Q14. Using integration by parts with u = ln x, dv = dx, the derivative du is:

  • dx
  • 1/x dx
  • ln x dx
  • x dx

Correct Answer: 1/x dx

Q15. For ∫ x^n e^{ax} dx, repeated integration by parts generates:

  • A polynomial times e^{ax} plus constant
  • A logarithmic expression
  • A rational function only
  • An indefinite divergent series

Correct Answer: A polynomial times e^{ax} plus constant

Q16. Integration by parts can derive reduction formulas. Which is a valid reduction relation for I_n = ∫ x^n e^{x} dx?

  • I_n = x^n e^{x} – n I_{n-1}
  • I_n = e^{x} – n I_{n-1}
  • I_n = n I_{n-1} – x^n e^{x}
  • I_n = x^{n-1} e^{x} + n I_{n-1}

Correct Answer: I_n = x^n e^{x} – n I_{n-1}

Q17. Evaluate ∫_0^{\pi/2} x cos x dx. Which boundary term appears after parts?

  • [x sin x]_0^{\pi/2}
  • [x cos x]_0^{\pi/2}
  • [cos x]_0^{\pi/2}

Correct Answer: [x sin x]_0^{\pi/2}

Q18. Using parts, ∫ arctan x dx is best started with u = arctan x and dv = dx. The resulting v is:

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

Correct Answer: x

Q19. For integrals of the form ∫ x ln(ax) dx, integration by parts yields which term?

  • x ln(ax) – x
  • ln(ax) – x
  • x^2 ln(ax)/2
  • x/(ln a)

Correct Answer: x ln(ax) – x

Q20. In drug release modeling, integrating rate expressions like ∫ t k e^{-kt} dt often requires:

  • Integration by parts or recognition as gamma moments
  • Partial fractions only
  • Trigonometric substitution
  • No integration, it’s algebraic

Correct Answer: Integration by parts or recognition as gamma moments

Q21. Which of the following is the correct result for ∫ x e^{2x} dx?

  • (x/2) e^{2x} – (1/4) e^{2x} + C
  • (x/2) e^{2x} + (1/4) e^{2x} + C
  • (x/2) e^{2x} – (1/2) e^{2x} + C
  • e^{2x}(x – 1)/2 + C

Correct Answer: (x/2) e^{2x} – (1/4) e^{2x} + C

Q22. For ∫ ln(x+1) dx, using parts with u = ln(x+1), dv = dx gives which antiderivative?

  • (x+1) ln(x+1) – x + C
  • x ln(x+1) – x + C
  • (x+1) ln(x+1) – (x+1) + C
  • ln(x+1) + C

Correct Answer: (x+1) ln(x+1) – (x+1) + C

Q23. When applying integration by parts to a definite integral, if v du term is easier, you should:

  • Always avoid parts
  • Choose u and dv to make v du simpler
  • Choose u as the more complicated function
  • Choose dv as the derivative of u

Correct Answer: Choose u and dv to make v du simpler

Q24. The integral ∫ x^2 ln x dx equals:

  • (x^3/3) ln x – x^3/9 + C
  • (x^3/3) ln x – x^3/3 + C
  • x^2 ln x – x^2/2 + C
  • (x^3/3) ln x + x^3/9 + C

Correct Answer: (x^3/3) ln x – x^3/9 + C

Q25. For ∫ e^{ax} cos(bx) dx, integration by parts twice yields an expression solvable by:

  • Algebraic elimination to isolate the integral
  • Substitution to exponential only
  • Trigonometric identities only
  • Partial fractions

Correct Answer: Algebraic elimination to isolate the integral

Q26. Applying integration by parts to ∫_0^{\infty} t e^{-t} dt helps compute:

  • Mean of exponential distribution
  • Median of distribution
  • Mode only
  • Differential equation solution only

Correct Answer: Mean of exponential distribution

Q27. Which is the correct antiderivative for ∫ x cos x dx?

  • x sin x + cos x + C
  • x sin x – cos x + C
  • sin x – x cos x + C
  • x cos x – sin x + C

Correct Answer: x sin x – cos x + C

Q28. For repeated integration by parts of ∫ x^n ln x dx, the general strategy is:

  • Differentiate ln x repeatedly
  • Differentiate x^n until zero while integrating ln x
  • Use substitution x = e^u
  • Use partial fractions

Correct Answer: Differentiate x^n until zero while integrating ln x

Q29. When integrating ∫ ln^2 x dx, using parts with u = ln^2 x gives du = 2 ln x (1/x) dx and helps reduce power by:

  • One power per application
  • Two powers per application
  • It increases powers
  • It does not change power

Correct Answer: One power per application

Q30. Integration by parts can be used to evaluate ∫ x/(1+x^2) dx by choosing u = x and dv = 1/(1+x^2) dx. The result simplifies to:

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

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

Q31. For ∫_0^1 ln x dx, integration by parts yields a finite value of:

  • -1
  • -1/2
  • 0
  • 1

Correct Answer: -1

Q32. Which integral representing a pharmacokinetic moment often uses integration by parts: AUMC = ∫ t C(t) dt is used to compute?

  • Area under concentration-time curve (AUC)
  • Area under the first moment curve (AUMC) for MRT calculation
  • Bioavailability only
  • Clearance directly

Correct Answer: Area under the first moment curve (AUMC) for MRT calculation

Q33. The integral ∫ e^{x} ln x dx is usually solved by choosing u = ln x, dv = e^{x} dx. The result involves:

  • e^{x} ln x – ∫ e^{x}/x dx
  • e^{x} ln x – ∫ e^{x} x dx
  • ln x e^{x} + C with no residual integral
  • Substitution only

Correct Answer: e^{x} ln x – ∫ e^{x}/x dx

Q34. Integration by parts is helpful for integrals involving inverse functions because:

  • It eliminates inverse functions immediately
  • It transfers differentiation from inverse to algebraic factor
  • It replaces inverse with exponential
  • It is not helpful for inverse functions

Correct Answer: It transfers differentiation from inverse to algebraic factor

Q35. Solve ∫ x sin x dx. Which is the correct antiderivative?

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

Correct Answer: -x cos x + sin x + C

Q36. If u = f(x) and dv = g'(x) dx, integration by parts requires computing v as:

  • An antiderivative of g'(x), i.e., g(x)
  • The derivative of g'(x)
  • g'(x) itself
  • f'(x)

Correct Answer: An antiderivative of g'(x), i.e., g(x)

Q37. Integration by parts can convert integrals into known forms. For ∫ x^2 e^{-x} dx, which step is correct first?

  • Take u = e^{-x}, dv = x^2 dx
  • Take u = x^2, dv = e^{-x} dx
  • Perform substitution x = -t
  • Use trigonometric identities

Correct Answer: Take u = x^2, dv = e^{-x} dx

Q38. Which of the following integrals yields a term involving the exponential integral Ei(x) after applying integration by parts?

  • ∫ e^{x}/x dx
  • ∫ x e^{x} dx
  • ∫ e^{2x} dx
  • ∫ x^2 e^{x} dx

Correct Answer: ∫ e^{x}/x dx

Q39. For integrals of the form ∫ P(x) ln Q(x) dx where P and Q are polynomials, integration by parts typically sets u =:

  • P(x)
  • ln Q(x)
  • Q(x)
  • 1/P(x)

Correct Answer: ln Q(x)

Q40. In deriving the formula for mean residence time (MRT) in a one-compartment model, integration by parts is used to relate AUMC and AUC because:

  • AUMC is a derivative of AUC
  • AUMC involves t*C(t) which is a product of time and concentration
  • AUC is always zero
  • MRT does not use integrals

Correct Answer: AUMC involves t*C(t) which is a product of time and concentration

Q41. Which choice of u is best for ∫ x^3 ln x dx?

  • u = x^3, dv = ln x dx
  • u = ln x, dv = x^3 dx
  • u = x, dv = x^2 ln x dx
  • u = 1, dv = x^3 ln x dx

Correct Answer: u = ln x, dv = x^3 dx

Q42. The integral ∫_0^{\infty} t^n e^{-t} dt is evaluated via repeated parts and equals:

  • n! (gamma function relation)
  • 1/n!
  • e^{-n}
  • Infinite

Correct Answer: n! (gamma function relation)

Q43. For ∫ x ln(1+x) dx, after parts the remaining integral involves:

  • ∫ x/(1+x) dx
  • ∫ ln(1+x) dx only
  • ∫ 1 dx only
  • ∫ x^2 dx

Correct Answer: ∫ x/(1+x) dx

Q44. Which statement about integration by parts and definite integrals is true?

  • Boundary terms always cancel for improper integrals
  • Boundary terms must be evaluated carefully and may require limits
  • Integration by parts cannot be used for improper integrals
  • Boundary terms are irrelevant for physical applications

Correct Answer: Boundary terms must be evaluated carefully and may require limits

Q45. Integration by parts applied to ∫ arccos x dx with u = arccos x yields an integral containing:

  • ∫ x / sqrt(1-x^2) dx
  • ∫ sqrt(1-x^2) dx
  • ∫ 1/(1+x^2) dx
  • ∫ ln x dx

Correct Answer: ∫ x / sqrt(1-x^2) dx

Q46. In pharmacokinetic modelling, integrating C(t) = C0 e^{-kt} from 0 to ∞ yields AUC = C0/k. This result can be obtained directly or by:

  • Integration by parts giving same result
  • Partial fractions only
  • Trigonometric substitution
  • Series expansion only

Correct Answer: Integration by parts giving same result

Q47. Which of these integrals simplifies directly by taking u = ln x in integration by parts?

  • ∫ ln x / x dx
  • ∫ x ln x dx
  • ∫ e^{x} ln x dx
  • ∫ ln^2 x dx

Correct Answer: ∫ x ln x dx

Q48. For ∫ x cosh x dx, integration by parts yields an antiderivative:

  • x sinh x – cosh x + C
  • x sinh x + cosh x + C
  • sinh x – x cosh x + C
  • x cosh x – sinh x + C

Correct Answer: x sinh x – cosh x + C

Q49. When integrating ∫ ln(sin x) dx by parts with u = ln(sin x), the differential du involves:

  • cot x dx
  • tan x dx
  • sec x dx
  • csc x dx

Correct Answer: cot x dx

Q50. Which practical skill does repeated practice of integration by parts most directly improve for B.Pharm students?

  • Ability to design clinical trials
  • Ability to symbolically integrate products and apply results to pharmacokinetics
  • Ability to prepare sterile formulations
  • Ability to perform wet lab titrations

Correct Answer: Ability to symbolically integrate products and apply results to pharmacokinetics

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