Integration by substitution by parts and by partial fractions - Question Bank

1. 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
2. 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
3. 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.
4. 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)
5. 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
6. 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
7. The method of substitution is essentially reversing the:
A) Chain Rule
B) Product Rule
C) Quotient Rule
D) Power Rule
8. 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
9. 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
10. 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)
11. 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
12. 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)
13. 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
14. 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
15. 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
16. 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
17. 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))
18. 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.
19. What is the integral of ∫ tan(x) dx?
A) -ln|cos(x)| + C
B) ln|sin(x)| + C
C) sec^2(x) + C
D) ln|sec(x)| + C
20. 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
21. What is the integral of ∫ x^2 dx?
A) x^3/3 + C
B) 2x + C
C) x^3 + C
D) 3x^2 + C
22. 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)
23. 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
24. 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
25. 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
26. 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
27. 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
28. 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
29. 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
30. 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
31. 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)
32. 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)
33. 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)
34. 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
35. 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
36. 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
37. 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
38. 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)
39. 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)
40. 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
41. 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.
42. 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.
43. 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.
44. A rational function is a ratio of two:
A) Polynomials.
B) Trigonometric functions.
C) Logarithmic functions.
D) Exponential functions.
45. Integration by partial fractions is primarily used for integrating:
A) Rational functions.
B) Irrational functions.
C) Trigonometric functions.
D) Exponential functions.
46. 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)
47. 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
48. 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.
49. Which rule is fundamental to integration by parts?
A) Chain Rule
B) Product Rule
C) Quotient Rule
D) Sum Rule
50. 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.
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