Rules of integration MCQs With Answer

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Rules of integration MCQs With Answer provide B.Pharm students with essential practice on integration techniques used in pharmacokinetics, drug release kinetics, and quantitative pharmaceutical calculations. This concise, keyword-rich introduction covers core integration rules—power rule, substitution, integration by parts, partial fractions, trigonometric and exponential integrals—and links them to real-world topics like area under the curve (AUC), clearance, half-life, and Higuchi models. Focused practice on Rules of integration MCQs With Answer helps students master analytical steps required for deriving concentration–time relationships and solving dosage problems. Clear explanations and targeted problems build competency for exams and research. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the integral of x^3 with respect to x?

  • 3x^2
  • x^4
  • 4x^3
  • 1/4 x^4

Correct Answer: 1/4 x^4

Q2. According to the constant multiple rule, ∫ 5 f(x) dx equals:

  • 5 + ∫ f(x) dx
  • 5 ∫ f(x) dx
  • ∫ f(5x) dx
  • ∫ f(x) dx / 5

Correct Answer: 5 ∫ f(x) dx

Q3. Using the sum rule, ∫ [x^2 + sin x] dx equals:

  • ∫ x^2 dx + ∫ sin x dx
  • ∫ x^2 dx * ∫ sin x dx
  • ∫ (x^2 sin x) dx
  • ∫ x^2 dx – ∫ sin x dx

Correct Answer: ∫ x^2 dx + ∫ sin x dx

Q4. The integral ∫ e^{ax} dx is:

  • e^{ax} / a
  • a e^{ax}
  • ln|e^{ax}|
  • e^{ax}

Correct Answer: e^{ax} / a

Q5. ∫ (1/x) dx for x>0 equals:

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

Correct Answer: ln|x| + C

Q6. Which substitution is most appropriate to integrate ∫ 2x e^{x^2} dx?

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

Correct Answer: u = x^2

Q7. The fundamental theorem of calculus states that d/dx ∫_a^x f(t) dt equals:

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

Correct Answer: f(x)

Q8. For a first-order elimination C(t)=C0 e^{-kt}, the definite integral ∫_0^∞ C(t) dt equals:

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

Correct Answer: C0 / k

Q9. Derive t1/2 from first-order kinetics using integration; t1/2 equals:

  • ln 2 / k
  • k / ln 2
  • 2 / k
  • ln k / 2

Correct Answer: ln 2 / k

Q10. Integration by parts formula is:

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

Correct Answer: ∫ u dv = uv – ∫ v du

Q11. ∫ ln x dx equals:

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

Correct Answer: x ln x – x + C

Q12. The integral ∫ cos(ax) dx is:

  • sin(ax)/a + C
  • cos(ax)/a + C
  • tan(ax)/a + C
  • -sin(ax)/a + C

Correct Answer: sin(ax)/a + C

Q13. ∫ sec^2 x dx equals:

  • tan x + C
  • sec x + C
  • ln|sec x + tan x| + C
  • -cot x + C

Correct Answer: tan x + C

Q14. Which method is best for ∫ (x+1)/(x^2+x) dx?

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

Correct Answer: Partial fractions

Q15. The improper integral ∫_1^∞ 1/x^p dx converges when:

  • p < 1
  • p = 1
  • p > 1
  • never converges

Correct Answer: p > 1

Q16. Change of limits under substitution u = g(x) requires:

  • keeping original x-limits
  • converting x-limits to u-limits
  • doubling the limits
  • neglecting limits entirely

Correct Answer: converting x-limits to u-limits

Q17. The integral ∫ sqrt(x) dx equals:

  • 2 x^{1/2} / 3 + C
  • 2/3 x^{3/2} + C
  • x^{3/2} + C
  • 3/2 x^{1/2} + C

Correct Answer: 2/3 x^{3/2} + C

Q18. ∫ 1/(x^2 + a^2) dx equals:

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

Correct Answer: 1/a arctan(x/a) + C

Q19. To integrate ∫ (2x)/(x^2+1) dx, use substitution u = x^2+1 to get:

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

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

Q20. The area under C(t)=k sqrt(t) from 0 to T uses which integral result?

  • k * 2/3 T^{3/2}
  • k * 3/2 T^{1/2}
  • k T
  • k ln T

Correct Answer: k * 2/3 T^{3/2}

Q21. The definite integral property ∫_a^b f(x) dx = -∫_b^a f(x) dx follows from:

  • linearity of integration
  • odd/even function symmetry
  • orientation of limits
  • integration by parts

Correct Answer: orientation of limits

Q22. ∫_0^1 1/√x dx converges and equals:

  • 2
  • 1
  • 0
  • diverges to ∞

Correct Answer: 2

Q23. Which integral is used to compute area under plasma concentration-time curve (AUC)?

  • ∫ C(t) dt
  • ∫ t dt
  • ∫ dC/dt dt
  • ∫ k dt

Correct Answer: ∫ C(t) dt

Q24. For IV bolus with clearance CL and dose D, AUC equals (assuming one-compartment first-order):

  • D * CL
  • D / CL
  • CL / D
  • D + CL

Correct Answer: D / CL

Q25. The integral ∫ t e^{-kt} dt from 0 to ∞ (with k>0) equals:

  • 1/k
  • 1/k^2
  • k
  • 0

Correct Answer: 1/k^2

Q26. AUMC (area under the first moment curve) uses which integrand?

  • t + C(t)
  • t * C(t)
  • C(t)/t
  • C'(t)

Correct Answer: t * C(t)

Q27. Mean residence time MRT is defined as AUMC/AUC. For a one-compartment IV bolus with first-order elimination MRT equals:

  • 1/k
  • k
  • ln 2 / k
  • D / CL

Correct Answer: 1/k

Q28. Integrate ∫ cos(2x) dx:

  • sin(2x)/2 + C
  • cos(2x)/2 + C
  • tan(2x)/2 + C
  • -sin(2x)/2 + C

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

Q29. The integral ∫ a^x dx (a>0, a≠1) equals:

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

Correct Answer: a^x / ln a + C

Q30. Is ∫ e^{-x^2} dx expressible in elementary functions?

  • Yes, always
  • No, it requires the error function (erf)
  • Yes, as ln function
  • Only for x rational

Correct Answer: No, it requires the error function (erf)

Q31. To evaluate ∫ (x^2-1)/(x^2-1) dx where denominator cancels, the integral equals:

  • ∫ 1 dx = x + C
  • ln(x^2-1) + C
  • 0
  • 1/(x^2-1) + C

Correct Answer: ∫ 1 dx = x + C

Q32. The integral ∫ (1/(x ln x)) dx equals:

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

Correct Answer: ln|ln x| + C

Q33. For zero-order absorption with rate k0, concentration increases linearly; integrating k0 dt gives:

  • k0 t + C
  • k0 / t + C
  • ln(k0 t) + C
  • e^{k0 t} + C

Correct Answer: k0 t + C

Q34. For partial fraction decomposition, ∫ dx/(x(x+1)) equals:

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

Correct Answer: ln|x| – ln|x+1| + C

Q35. Which integral technique is best to integrate polynomial times exponential like ∫ x e^{ax} dx?

  • Partial fractions
  • Integration by parts
  • Trigonometric substitution
  • Direct power rule

Correct Answer: Integration by parts

Q36. The definite integral ∫_0^T C0 e^{-kt} dt equals C0(1 – e^{-kT})/k. This expresses:

  • finite AUC from 0 to T
  • instantaneous concentration
  • peak concentration only
  • clearance directly

Correct Answer: finite AUC from 0 to T

Q37. Use substitution u = ln x to transform ∫ dx/x into:

  • ∫ du
  • ∫ u du
  • ∫ e^u du
  • ∫ ln u du

Correct Answer: ∫ du

Q38. The integral ∫ 1/(1+x^2) dx is commonly used in PK and equals:

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

Correct Answer: arctan x + C

Q39. To integrate ∫ (x^2+1)/(x^3+x) dx simplify first by:

  • polynomial long division and partial fractions
  • trigonometric substitution
  • integration by parts directly
  • numerical integration only

Correct Answer: polynomial long division and partial fractions

Q40. The derivative of ∫_a^{g(x)} f(t) dt equals:

  • f(a) g'(x)
  • f(g(x)) g'(x)
  • ∫ f(t) dt
  • g(x) f'(g(x))

Correct Answer: f(g(x)) g'(x)

Q41. When integrating rational functions where denominator is irreducible quadratic, expect an arctan term after:

  • trigonometric substitution only
  • completing the square
  • integration by parts
  • ignoring the quadratic

Correct Answer: completing the square

Q42. For concentration C(t)=C0/(1+kt) (a simple saturable model), ∫ C(t) dt involves:

  • ln(1+kt) term
  • power law only
  • trigonometric functions
  • no elementary antiderivative

Correct Answer: ln(1+kt) term

Q43. ∫_0^∞ x^{n} e^{-x} dx equals n! for which n values (gamma function relation)?

  • integer n ≥ 0
  • only n = 0
  • no integer values
  • negative integers only

Correct Answer: integer n ≥ 0

Q44. To compute time to reach concentration fraction C/C0 in first-order kinetics, integrating dC/C = -k dt gives:

  • t = ln(C/C0)/k
  • t = -ln(C/C0)/k
  • t = k ln(C/C0)
  • t = C0 / k

Correct Answer: t = -ln(C/C0)/k

Q45. The substitution for ∫ dx/(x^2-1) uses partial fractions resulting in:

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

Correct Answer: 1/2 ln|x-1| – 1/2 ln|x+1| + C

Q46. The integral ∫ (cos x)^2 dx can be simplified using which identity?

  • double-angle identity cos^2 x = (1+cos 2x)/2
  • product-to-sum of sines only
  • no identity simplifies it
  • tan double-angle

Correct Answer: double-angle identity cos^2 x = (1+cos 2x)/2

Q47. To find AUC numerically from discrete concentration points, which rule is based on definite integral approximations?

  • Trapezoidal rule
  • Simpson’s paradox
  • Laplace transform
  • Integration by parts

Correct Answer: Trapezoidal rule

Q48. The integral ∫ (1/(x^2 – a^2)) dx results in logarithms after partial fractions as:

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

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

Q49. Which integral property helps split an AUC over two intervals [a,c] and [c,b]?

  • Additivity: ∫_a^b = ∫_a^c + ∫_c^b
  • Multiplicativity
  • Subtraction only
  • No such property exists

Correct Answer: Additivity: ∫_a^b = ∫_a^c + ∫_c^b

Q50. For practical pharmacokinetic modeling, which integration skill is most essential?

  • Recognizing when to apply substitution, parts, or partial fractions to obtain analytic AUC and moments
  • Only numerical integration without analytic understanding
  • Only memorizing integral tables
  • Avoiding integration and using heuristics

Correct Answer: Recognizing when to apply substitution, parts, or partial fractions to obtain analytic AUC and moments

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