Mastering the standard formulae of integration is essential for B. Pharm students tackling calculus in pharmaceutical analysis, pharmacokinetics and formulation modeling. This concise, informative collection of MCQs focuses on key integration formulas — power rule, exponential and logarithmic integrals, trigonometric and inverse-trigonometric integrals, definite vs. indefinite integrals, substitution, integration by parts, partial fractions and reduction formulas — with clear answers and practical problem framing relevant to pharmacy coursework. Each question reinforces technique selection, algebraic manipulation and application of limits when evaluating definite integrals. Use these targeted MCQs to strengthen calculation speed and conceptual understanding before exams and lab work. Now let’s test your knowledge with 50 MCQs on this topic.
Q1. ∫ x^3 dx
- x^4/4 + C
- x^3/3 + C
- 3x^4 + C
- ln|x| + C
Correct Answer: x^4/4 + C
Q2. ∫ 1/x dx
- ln|x| + C
- 1/x + C
- x + C
- e^x + C
Correct Answer: ln|x| + C
Q3. ∫ e^(2x) dx
- (1/2) e^(2x) + C
- 2 e^(2x) + C
- e^(2x) + C
- ln|e^(2x)| + C
Correct Answer: (1/2) e^(2x) + C
Q4. ∫ sin x dx
- -cos x + C
- cos x + C
- tan x + C
- sin x + C
Correct Answer: -cos x + C
Q5. ∫ cos x dx
- sin x + C
- -sin x + C
- cos x + C
- ln|cos x| + C
Correct Answer: sin x + C
Q6. ∫ sec^2 x dx
- tan x + C
- sec x + C
- ln|sec x| + C
- cot x + C
Correct Answer: tan x + C
Q7. ∫ 1/(1+x^2) dx
- arctan x + C
- arcsin x + C
- ln|1+x^2| + C
- 1/(1+x^2) + C
Correct Answer: arctan x + C
Q8. ∫ 1/√(1-x^2) dx
- arcsin x + C
- arccos x + C
- ln|x + √(1-x^2)| + C
- 1/√(1-x^2) + C
Correct Answer: arcsin x + C
Q9. ∫ x e^x dx
- (x – 1) e^x + C
- x e^x + C
- (1 – x) e^x + C
- e^x + C
Correct Answer: (x – 1) e^x + C
Q10. ∫ ln x dx
- x ln x – x + C
- ln^2 x + C
- x ln x + C
- 1/x + C
Correct Answer: x ln x – x + C
Q11. ∫ a^x dx (a>0, a≠1)
- a^x / ln a + C
- ln a * a^x + C
- a^x + C
- ln|a^x| + C
Correct Answer: a^x / ln a + C
Q12. ∫ cos^2 x dx
- (x/2) + (sin 2x)/4 + C
- (x/2) – (sin 2x)/4 + C
- sin x cos x + C
- cos x + C
Correct Answer: (x/2) + (sin 2x)/4 + C
Q13. ∫ sin^2 x dx
- (x/2) – (sin 2x)/4 + C
- (x/2) + (sin 2x)/4 + C
- -cos x + C
- sin x + C
Correct Answer: (x/2) – (sin 2x)/4 + C
Q14. ∫ dx/(x^2 + a^2)
- (1/a) arctan(x/a) + C
- arcsin(x/a) + C
- ln|x^2 + a^2| + C
- 1/(x^2 + a^2) + C
Correct Answer: (1/a) arctan(x/a) + C
Q15. ∫ dx/√(x^2 + a^2)
- ln|x + √(x^2 + a^2)| + C
- arcsinh(x/a) + C
- arctan(x/a) + C
- √(x^2 + a^2) + C
Correct Answer: ln|x + √(x^2 + a^2)| + C
Q16. ∫ dx/(x^2 – a^2)
- (1/2a) ln|(x – a)/(x + a)| + C
- arctan(x/a) + C
- ln|x^2 – a^2| + C
- 1/(x^2 – a^2) + C
Correct Answer: (1/2a) ln|(x – a)/(x + a)| + C
Q17. ∫ (2x)/(x^2 + 1) dx
- ln(x^2 + 1) + C
- x^2 + C
- 1/(x^2 + 1) + C
- arctan x + C
Correct Answer: ln(x^2 + 1) + C
Q18. ∫ 1/√x dx
- 2 √x + C
- √x + C
- ln x + C
- 1/(2 √x) + C
Correct Answer: 2 √x + C
Q19. ∫ x^(-2) dx
- -1/x + C
- 1/x + C
- ln|x| + C
- x^-1/(-1) + C
Correct Answer: -1/x + C
Q20. ∫ tan x dx
- -ln|cos x| + C
- ln|cos x| + C
- ln|sin x| + C
- sec x + C
Correct Answer: -ln|cos x| + C
Q21. ∫ cot x dx
- ln|sin x| + C
- -ln|sin x| + C
- ln|cos x| + C
- -ln|cos x| + C
Correct Answer: ln|sin x| + C
Q22. ∫ sec x tan x dx
- sec x + C
- tan x + C
- ln|sec x| + C
- sec^2 x + C
Correct Answer: sec x + C
Q23. ∫ csc x cot x dx
- -csc x + C
- csc x + C
- ln|csc x| + C
- cot x + C
Correct Answer: -csc x + C
Q24. ∫ x^2/(x^3 + 1) dx
- (1/3) ln(x^3 + 1) + C
- ln(x^2 + 1) + C
- x/(x^3 + 1) + C
- 1/(x^3 + 1) + C
Correct Answer: (1/3) ln(x^3 + 1) + C
Q25. ∫ (2x+1)/(x^2 + x) dx
- ln|x(x+1)| + C
- ln|x^2 + x| + C
- 1/(x^2 + x) + C
- x + ln|x+1| + C
Correct Answer: ln|x(x+1)| + C
Q26. ∫_0^1 x dx
- 1/2
- 1
- 1/3
- 0
Correct Answer: 1/2
Q27. ∫_0^π sin x dx
- 2
- 0
- 1
- π
Correct Answer: 2
Q28. ∫_0^{π/2} cos^2 x dx
- π/4
- π/2
- 1
- 0
Correct Answer: π/4
Q29. ∫ e^x sin x dx
- (e^x/2)(sin x – cos x) + C
- e^x(sin x + cos x) + C
- (e^x)(sin x) + C
- (e^x/2)(sin x + cos x) + C
Correct Answer: (e^x/2)(sin x – cos x) + C
Q30. ∫ e^{ax} sin(bx) dx (general formula)
- e^{ax}(a sin bx – b cos bx)/(a^2 + b^2) + C
- e^{ax}(b sin bx + a cos bx)/(a^2 + b^2) + C
- e^{ax} sin(bx)/(a+b) + C
- e^{ax}(a cos bx + b sin bx)/(a^2 – b^2) + C
Correct Answer: e^{ax}(a sin bx – b cos bx)/(a^2 + b^2) + C
Q31. ∫ dx/(x √(x^2 – 1))
- arcsec|x| + C
- arccos|1/x| + C
- arcsin|1/x| + C
- ln|x + √(x^2 -1)| + C
Correct Answer: arcsec|x| + C
Q32. ∫ arctan x dx
- x arctan x – (1/2) ln(1 + x^2) + C
- arctan x + C
- ln(1 + x^2) + C
- x ln(1 + x^2) + C
Correct Answer: x arctan x – (1/2) ln(1 + x^2) + C
Q33. ∫ arcsin x dx
- x arcsin x + √(1 – x^2) + C
- x arcsin x – √(1 – x^2) + C
- arcsin x + C
- ln|x + √(1 – x^2)| + C
Correct Answer: x arcsin x + √(1 – x^2) + C
Q34. ∫ x/(x^2 + 1) dx
- (1/2) ln(x^2 + 1) + C
- ln(x^2 + 1) + C
- arctan x + C
- 1/(x^2 + 1) + C
Correct Answer: (1/2) ln(x^2 + 1) + C
Q35. ∫ x^2/(x^2 + 1) dx
- x – arctan x + C
- x + arctan x + C
- ln(x^2 + 1) + C
- 1 + C
Correct Answer: x – arctan x + C
Q36. ∫ dx/(x^2 + 2x + 2)
- arctan(x + 1) + C
- arctan(x/√2) + C
- ln|x^2 + 2x + 2| + C
- 1/(x^2 + 2x + 2) + C
Correct Answer: arctan(x + 1) + C
Q37. Reduction formula for ∫ sin^n x dx is
- ∫ sin^n x dx = – (1/n) sin^{n-1} x cos x + ((n-1)/n) ∫ sin^{n-2} x dx + C
- ∫ sin^n x dx = (1/n) sin^{n-1} x cos x + ((n-1)/n) ∫ sin^{n-2} x dx + C
- ∫ sin^n x dx = sin^{n+1} x/(n+1) + C
- ∫ sin^n x dx = -cos x sin^{n} x + C
Correct Answer: ∫ sin^n x dx = – (1/n) sin^{n-1} x cos x + ((n-1)/n) ∫ sin^{n-2} x dx + C
Q38. Integration by parts formula is
- ∫ u dv = u v – ∫ v du
- ∫ u dv = u v + ∫ v du
- ∫ u dv = (du)(dv)
- ∫ u dv = u’ v’ + C
Correct Answer: ∫ u dv = u v – ∫ v du
Q39. ∫ (1 + tan^2 x) dx
- tan x + C
- sec x + C
- ln|sec x| + C
- cot x + C
Correct Answer: tan x + C
Q40. ∫ sec x dx
- ln|sec x + tan x| + C
- ln|sec x| + C
- sec x tan x + C
- tan x + C
Correct Answer: ln|sec x + tan x| + C
Q41. ∫ dx/(a^2 – x^2)
- (1/(2a)) ln|(a + x)/(a – x)| + C
- arctan(x/a) + C
- 1/(a^2 – x^2) + C
- ln|a^2 – x^2| + C
Correct Answer: (1/(2a)) ln|(a + x)/(a – x)| + C
Q42. ∫ cos x / sin^2 x dx
- -csc x + C
- csc x + C
- ln|sin x| + C
- 1/sin x + C
Correct Answer: -csc x + C
Q43. ∫ dx/√(a^2 – x^2)
- arcsin(x/a) + C
- arccos(x/a) + C
- ln|x + √(a^2 – x^2)| + C
- 1/√(a^2 – x^2) + C
Correct Answer: arcsin(x/a) + C
Q44. ∫ x √(x^2 + 1) dx
- (1/3)(x^2 + 1)^{3/2} + C
- (1/2)(x^2 + 1)^{1/2} + C
- √(x^2 + 1) + C
- x(x^2 + 1)^{1/2} + C
Correct Answer: (1/3)(x^2 + 1)^{3/2} + C
Q45. ∫_0^1 ln x dx
- -1
- 0
- 1
- ln 1
Correct Answer: -1
Q46. ∫ x / √(1 – x^2) dx
- -√(1 – x^2) + C
- √(1 – x^2) + C
- arcsin x + C
- ln|1 – x^2| + C
Correct Answer: -√(1 – x^2) + C
Q47. ∫ dx/(x ln x)
- ln|ln x| + C
- ln|x| + C
- 1/ln x + C
- ln|x ln x| + C
Correct Answer: ln|ln x| + C
Q48. ∫ cos(ax) dx
- (1/a) sin(ax) + C
- a sin(ax) + C
- sin(ax) + C
- ln|cos(ax)| + C
Correct Answer: (1/a) sin(ax) + C
Q49. ∫ sin^3 x dx
- -cos x + (1/3) cos^3 x + C
- cos x – (1/3) cos^3 x + C
- -(1/3) cos^3 x + C
- sin^2 x/2 + C
Correct Answer: -cos x + (1/3) cos^3 x + C
Q50. ∫ 1/(x^2 + x) dx
- ln|x/(x + 1)| + C
- ln|x^2 + x| + C
- 1/(x + 1) + C
- ln|x| + C
Correct Answer: ln|x/(x + 1)| + C
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