Advanced Integration Techniques

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Question 1

Partial fractions can be applied directly only when:

  • Numerator degree ≥ denominator degree

  • Numerator degree < denominator degree

  • Numerator degree = denominator degree

  • Denominator is linear

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Question 2

If denominator has an irreducible quadratic factor, the correct term is:

  • A/Quadratic

  • Ax + B/ Quadratic

  • A/(Quadratic)2

  • Only linear terms are used

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Question 3

The method of partial fractions is used when the integrand is:

  • A trigonometric function

  • An exponential function

  • A rational function

  • A radical (root) function

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Question 4

Which is the correct first step if the given rational function’s denominator cannot be factored into linear factors over real numbers (e.g. irreducible quadratic)?

  • Immediately apply simple partial fractions form for linear factors

  • Use standard integrals for quadratic denominators or complete the square

  • Multiply numerator and denominator by something

  • Convert into trigonometric integral

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Question 5

If you have an “improper” rational function (degree of numerator ≥ degree of denominator), you should first:

  • Directly apply partial fractions

  • Use integration by parts

  • Use substitution method

  • Perform polynomial long division

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Question 6

Integration by Parts is generally used when the integrand is:

  • A composite function like f(g(x))

  • A product of two (or more) functions

  • A rational function (polynomial over polynomial)

  • A pure trigonometric function only

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Question 7

According to the rule of thumb for choosing uuu and dvdvdv, which mnemonic is used?

  • SOHCAHTOA

  • FOIL

  • ILATE

  • PEMDAS

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Question 8

In integration by parts, why is the choice of u and dv significant?

  • Because only u gets integrated

  • Because you must differentiate dv

  • Because differentiating u should simplify it and integrating dv should be manageable

  • Because you cannot exchange u and dv

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Question 9

What is the origin of the Integration by Parts formula?

  • It’s guessed from integral tables

  • It comes from reversing the product rule for differentiation

  • It is derived from the chain rule

  • It’s a special case of substitution

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Question 10

Which of the following is the correct integration of[Tex] \int x e^{x} dx[/Tex] using integration by parts?

  • exx2 /2 + C

  • ex(x − 1) + C

  • ex(x + 1) + C

  • xex + C

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