JEE Main - Integration by substitution by parts and by partial fractions Online Test (MCQ Quiz)

1. What is the primary technique used in integration by substitution?

A) Replacing a function with a simpler one using a new variable. B) Expressing the integrand as a sum of simpler rational functions. C) Using the product rule in reverse. D) Splitting the integral into two parts based on a product.

2. Which rule is fundamental to integration by parts?

A) Chain Rule B) Product Rule C) Quotient Rule D) Sum Rule

3. The formula for integration by parts is derived from:

A) The derivative of a product of two functions. B) The derivative of a quotient of two functions. C) The integral of a sum of functions. D) The integral of a composite function.

4. What is the standard formula for integration by parts?

A) ∫ u dv = uv - ∫ v du B) ∫ u dv = uv + ∫ v du C) ∫ u dv = u/v - ∫ v/u du D) ∫ u dv = v/u - ∫ u/v du

5. In integration by parts, the choice of 'u' and 'dv' is crucial. Which mnemonic is often used to decide which function to choose as 'u'?

A) LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) B) SOHCAHTOA (Trigonometric Ratios) C) PEMDAS (Order of Operations) D) FOIL (First, Outer, Inner, Last)

6. Integration by partial fractions is primarily used for integrating:

A) Rational functions. B) Irrational functions. C) Trigonometric functions. D) Exponential functions.

7. A rational function is a ratio of two:

A) Polynomials. B) Trigonometric functions. C) Logarithmic functions. D) Exponential functions.

8. What is the first step in integrating a rational function using partial fractions?

A) Decompose the rational function into simpler fractions. B) Differentiate the rational function. C) Integrate the numerator and denominator separately. D) Apply integration by parts.

9. If the degree of the numerator is greater than or equal to the degree of the denominator in a rational function, what is the first step before partial fraction decomposition?

A) Perform polynomial long division. B) Differentiate the numerator. C) Integrate the denominator. D) Apply trigonometric substitution.

10. Which type of factor in the denominator leads to a partial fraction of the form A/(ax+b)?

A) Linear factor. B) Quadratic factor. C) Repeated linear factor. D) Irreducible quadratic factor.

11. For a repeated linear factor (ax+b)^n in the denominator, the partial fraction decomposition includes terms up to:

A) A_n / (ax+b)^n B) A / (ax+b) C) A / (ax+b)^2 D) A / (ax+b)^n

12. An irreducible quadratic factor in the denominator (ax^2 + bx + c) leads to a partial fraction of the form:

A) (Ax + B) / (ax^2 + bx + c) B) A / (ax^2 + bx + c) C) B / (ax + c) D) (Ax + B) / (bx + c)

13. For a repeated irreducible quadratic factor (ax^2 + bx + c)^n, the partial fraction decomposition includes terms of the form:

A) (A_1x + B_1)/(ax^2+bx+c) + ... + (A_nx + B_n)/(ax^2+bx+c)^n B) A/(ax^2+bx+c) + ... + A_n/(ax^2+bx+c)^n C) (Ax+B)/(ax^2+bx+c) + (Cx+D)/(ax^2+bx+c)^2 D) A/(ax^2+bx+c)

14. What is the integral of x*e^x dx using integration by parts?

A) x*e^x - e^x + C B) x*e^x + e^x + C C) e^x - x*e^x + C D) e^x + x*e^x + C

15. Evaluate ∫ ln(x) dx using integration by parts.

A) x*ln(x) - x + C B) x*ln(x) + x + C C) ln(x) - x + C D) x - ln(x) + C

16. What is the result of ∫ x*sin(x) dx?

A) -x*cos(x) + sin(x) + C B) x*cos(x) - sin(x) + C C) -x*cos(x) - sin(x) + C D) x*sin(x) - cos(x) + C

17. Using substitution, what is the integral of 2x/(1+x^2) dx?

A) ln(1+x^2) + C B) 2*ln(1+x^2) + C C) arctan(x) + C D) log(1+x^2) + C

18. What substitution is useful for integrating functions of the form √(a^2 - x^2)?

A) x = a*sin(θ) B) x = a*tan(θ) C) x = a*sec(θ) D) x = a*cos(θ)

19. What substitution is useful for integrating functions of the form √(a^2 + x^2)?

A) x = a*tan(θ) B) x = a*sin(θ) C) x = a*sec(θ) D) x = a*cos(θ)

20. What substitution is useful for integrating functions of the form √(x^2 - a^2)?

A) x = a*sec(θ) B) x = a*sin(θ) C) x = a*tan(θ) D) x = a*cos(θ)

21. Evaluate ∫ (1 / (x^2 - 1)) dx using partial fractions.

A) (1/2) * ln|(x-1)/(x+1)| + C B) (1/2) * ln|(x+1)/(x-1)| + C C) ln|(x-1)/(x+1)| + C D) ln|(x+1)/(x-1)| + C

22. What is the integral of 1/(x^2 + a^2) dx?

A) (1/a) * arctan(x/a) + C B) arctan(x/a) + C C) (1/a) * arctan(x) + C D) a * arctan(x/a) + C

23. If ∫ f(x) dx = G(x) + C, and we use substitution x = g(t), then ∫ f(g(t))g'(t) dt is equal to:

A) G(g(t)) + C B) G(t) + C C) G(g(x)) + C D) f(g(t)) + C

24. Which integral can be solved using integration by parts?

A) ∫ x*cos(x) dx B) ∫ sin(x) dx C) ∫ 1/(x^2+1) dx D) ∫ tan(x) dx

25. Which integral is best suited for integration by partial fractions?

A) ∫ (2x+1) / (x^2-1) dx B) ∫ x*e^x dx C) ∫ ln(x) dx D) ∫ sin(x)*cos(x) dx

26. Consider the integral ∫ x^2 * e^x dx. How many times would integration by parts need to be applied?

A) 2 B) 1 C) 3 D) 0

27. What is the derivative of u in the integration by parts formula ∫ u dv = uv - ∫ v du?

A) du B) u C) dv D) v

28. What is the integral of dv in the integration by parts formula ∫ u dv = uv - ∫ v du?

A) v B) dv C) u D) du

29. If we choose u = x and dv = e^(-x) dx for ∫ x*e^(-x) dx, what is du and v?

A) du = dx, v = -e^(-x) B) du = x dx, v = e^(-x) C) du = dx, v = e^(-x) D) du = -dx, v = -e^(-x)

30. What is the integral of ∫ x^2 dx?

A) x^3/3 + C B) 2x + C C) x^3 + C D) 3x^2 + C

31. Using substitution t = x+1, evaluate ∫ (x+2)/(x+1) dx.

A) (x+1) - ln|x+1| + C B) (x+1) + ln|x+1| + C C) x + ln|x+1| + C D) ln|x+1| - (x+1) + C

32. What is the integral of ∫ tan(x) dx?

33. For the integral ∫ (x^2 + 3x + 2) / (x - 1) dx, what is the first step?

A) Perform polynomial long division. B) Use substitution. C) Apply integration by parts. D) Decompose into partial fractions.

34. What is the partial fraction decomposition of 1 / (x(x+1))?

A) 1/x - 1/(x+1) B) 1/(x+1) - 1/x C) 1/x + 1/(x+1) D) 1 / (x(x+1))

35. Evaluate ∫ (1 / (x^2 + 4)) dx.

A) (1/2) * arctan(x/2) + C B) arctan(x/2) + C C) (1/4) * arctan(x/2) + C D) 2 * arctan(x/2) + C

36. Which method is most suitable for ∫ e^x * sin(x) dx?

A) Integration by parts (applied twice) B) Substitution C) Partial fractions D) Trigonometric substitution

37. If x = tan(θ), then dx = ?

A) sec^2(θ) dθ B) tan^2(θ) dθ C) sec(θ)tan(θ) dθ D) cos^2(θ) dθ

38. What is the integral of ∫ x / (x^2 + 1) dx?

A) (1/2) * ln(x^2 + 1) + C B) ln(x^2 + 1) + C C) arctan(x) + C D) 2 * ln(x^2 + 1) + C

39. Consider ∫ (x^2) / (x - 1) dx. After polynomial division, the integrand becomes:

A) x + 1 + 1/(x-1) B) x + 1 - 1/(x-1) C) x - 1 + 1/(x-1) D) x - 1 - 1/(x-1)

40. What is the integral of ∫ x * cos(x^2) dx using substitution?

A) (1/2) * sin(x^2) + C B) sin(x^2) + C C) 2 * sin(x^2) + C D) cos(x^2) + C

41. For the integral ∫ (3x + 5) / ((x-1)(x+2)) dx, the partial fraction decomposition will be of the form:

A) A/(x-1) + B/(x+2) B) A/(x-1) + B/(x+2)^2 C) (Ax+B)/(x-1) + C/(x+2) D) A/(x-1)^2 + B/(x+2)

42. If ∫ f(x) dx = F(x) + C, then ∫ f(ax+b) dx = ?

A) (1/a) * F(ax+b) + C B) a * F(ax+b) + C C) F(ax+b) + C D) F(x/a + b) + C

43. What is the integral of ∫ x * arctan(x) dx?

A) (x^2/2)*arctan(x) - (1/2)*x + (1/2)*arctan(x) + C B) (x^2/2)*arctan(x) + (1/2)*x + (1/2)*arctan(x) + C C) x*arctan(x) - (1/2)*x^2 + C D) (x^2/2)*arctan(x) - x + C

44. The method of substitution is essentially reversing the:

A) Chain Rule B) Product Rule C) Quotient Rule D) Power Rule

45. Which integral requires integration by parts due to the product of a polynomial and an exponential function?

A) ∫ x*e^(3x) dx B) ∫ sin(3x) dx C) ∫ 1/(x+1) dx D) ∫ cos(x^2) dx

46. What is the integral of ∫ (2x+3) / (x^2+3x+5) dx?

A) ln(x^2+3x+5) + C B) 2*ln(x^2+3x+5) + C C) (1/2)*ln(x^2+3x+5) + C D) arctan(x^2+3x+5) + C

47. Consider the integral ∫ x^3 dx. Which integration technique is NOT needed?

A) Integration by parts B) Power rule for integration C) Substitution D) None of the above (direct integration is sufficient)

48. For the integral ∫ (x^2 + 1) / (x^2 - 1) dx, the first step is:

A) Polynomial long division. B) Partial fraction decomposition. C) Trigonometric substitution. D) Integration by parts.

49. What is the integral of ∫ e^x * (x+1) dx?

A) x*e^x + C B) (x+1)*e^x + C C) e^x + x*e^x + C D) e^x*(x+2) + C

50. Which method is most appropriate for ∫ (x+1) / (x^2+2x+2) dx?

A) Substitution (let u = x^2+2x+2) B) Integration by parts C) Partial fractions D) Trigonometric substitution

Integration by substitution by parts and by partial fractions - Online Test

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