AP Calculus BC Flashcards: Using Linear Partial Fractions

Study Using Linear Partial Fractions in AP Calculus BC with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus BC

Using Linear Partial Fractions

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QUESTION

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Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$ .

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ANSWER

$B = -4$ . Set $x=-1$ to isolate $B$ .

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What this deck covers

This deck focuses on Using Linear Partial Fractions, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$ .

Answer: $B = -4$ . Set $x=-1$ to isolate $B$ .

Flashcard 2: What is the result of using partial fractions on $\frac{1}{(x+1)^2}$ ?

Answer: $\frac{A}{x+1} + \frac{B}{(x+1)^2}$ . Repeated factor already factored, so direct decomposition.

Flashcard 3: Determine the partial fractions for $\frac{8x}{x^2-2x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-2}$ . Factor $x^2-2x = x(x-2)$ .

Flashcard 4: Determine the partial fractions for $\frac{8x}{x^2-2x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-2}$ . Factor $x^2-2x = x(x-2)$ .

Flashcard 5: What is the general form of a linear partial fraction decomposition?

Answer: $\frac{A}{x-a} + \frac{B}{x-b}$ . For distinct linear factors, each gets one term.

Flashcard 6: What is the first step in decomposing a rational function into partial fractions?

Answer: Factor the denominator. Essential for setting up partial fraction terms.

Flashcard 7: What is the first step in decomposing a rational function into partial fractions?

Answer: Factor the denominator. Essential for setting up partial fraction terms.

Flashcard 8: Identify the partial fractions for $\frac{7x+9}{x^2+x}$ .

Answer: $\frac{A}{x} + \frac{B}{x+1}$ . Factor $x^2+x = x(x+1)$ .

Flashcard 9: Find $A$ in the decomposition $\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}$ .

Answer: $A = 2$ . Set $x=0$ to find $A$ .

Flashcard 10: What is the denominator structure required for partial fraction decomposition?

Answer: Product of linear or irreducible quadratic factors. Standard form required for partial fraction method.

Flashcard 11: What is the purpose of equating coefficients in partial fraction decomposition?

Answer: To solve for unknown constants. Matching coefficients determines constants.

Flashcard 12: What is the advantage of using partial fractions in integration?

Answer: Simplifies the integral into manageable parts. Each simple fraction integrates easily.

Flashcard 13: Determine the partial fractions for $\frac{7x+4}{x^2-1}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x+1}$ . Factor $x^2-1 = (x-1)(x+1)$ .

Flashcard 14: What does each term in a partial fraction decomposition represent?

Answer: A simpler fraction with a linear denominator. Each term has constant numerator and linear denominator.

Flashcard 15: Find $A$ in the decomposition $\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}$ .

Answer: $A = 2$ . Set $x=0$ to find $A$ .

Flashcard 16: What is the partial fraction form for a simple linear factor like $x-a$ ?

Answer: $\frac{A}{x-a}$ . Each linear factor gets one partial fraction term.

Flashcard 17: Find $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}$ .

Answer: $B = 5$ . Set $x=-2$ to isolate $B$ .

Flashcard 18: Determine $A$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}$ .

Answer: $A = 3$ . Set $x=0$ to isolate $A$ .

Flashcard 19: What does each term in a partial fraction decomposition represent?

Answer: A simpler fraction with a linear denominator. Each term has constant numerator and linear denominator.

Flashcard 20: Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$ .

Answer: $B = -4$ . Set $x=-1$ to isolate $B$ .

Flashcard 21: Determine the partial fractions for $\frac{3x^2}{x^3-x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$ . Factor $x^3-x = x(x-1)(x+1)$ .

Flashcard 22: What is the result of using partial fractions on $\frac{1}{(x+1)^2}$ ?

Answer: $\frac{A}{x+1} + \frac{B}{(x+1)^2}$ . Repeated factor already factored, so direct decomposition.

Flashcard 23: Identify the partial fraction decomposition for $\frac{1}{x(x+1)}$ .

Answer: $\frac{A}{x} + \frac{B}{x+1}$ . Factor denominator $x(x+1)$ , then assign constants.

Flashcard 24: What must be true about the degree of the numerator relative to the denominator?

Answer: Degree of numerator < degree of denominator. Proper fractions are required for decomposition.

Flashcard 25: Find the partial fractions for $\frac{x+6}{x^2-5x+6}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x-3}$ . Factor $x^2-5x+6 = (x-2)(x-3)$ .

Flashcard 26: Identify the partial fraction form for $\frac{1}{(x-1)(x-2)(x-3)}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}$ . Three distinct linear factors need three terms.

Flashcard 27: Find the partial fraction decomposition of $\frac{4x+2}{x^2-4}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x+2}$ . Factor $x^2-4 = (x-2)(x+2)$ .

Flashcard 28: What is the form of partial fractions for a factor like $(x-a)^n$ ?

Answer: $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$ . Each power of repeated factor needs its own term.

Flashcard 29: What substitution is used to solve for $A$ in partial fractions?

Answer: Set $x$ to make other term zero. Strategic substitution eliminates other terms.

Flashcard 30: Identify the partial fraction form for $\frac{1}{(x-1)(x-2)(x-3)}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}$ . Three distinct linear factors need three terms.

Flashcard 31: What is the purpose of partial fraction decomposition in calculus?

Answer: To simplify integration or differentiation. Converts complex fractions into simpler integrable forms.

Flashcard 32: Identify the partial fractions for $\frac{2x+1}{x^2+x-6}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x+3}$ . Factor $x^2+x-6 = (x-2)(x+3)$ .

Flashcard 33: Identify the partial decomposition for $\frac{3x+1}{(x-3)^2}$ .

Answer: $\frac{A}{x-3} + \frac{B}{(x-3)^2}$ . Repeated factor requires two terms.

Flashcard 34: Identify the partial fractions for $\frac{5x}{x^2+3x}$ .

Answer: $\frac{A}{x} + \frac{B}{x+3}$ . Factor out $x$ from denominator first.

Flashcard 35: What is the form of partial fractions for a repeated linear factor like $(x-a)^2$ ?

Answer: $\frac{A}{x-a} + \frac{B}{(x-a)^2}$ . Repeated factors need terms for each power.

Flashcard 36: State the partial fraction form for $\frac{6x+9}{x^3(x-1)}$ .

Answer: $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}$ . $x^3$ factor creates three terms plus linear factor.

Flashcard 37: What is the partial fraction form for a simple linear factor like $x-a$ ?

Answer: $\frac{A}{x-a}$ . Each linear factor gets one partial fraction term.

Flashcard 38: What must be done if the rational function is improper for partial fractions?

Answer: Perform polynomial long division first. Convert improper to proper fraction first.

Flashcard 39: What is a key benefit of partial fraction decomposition for solving integrals?

Answer: Breaks down complex rational functions. Makes integration much simpler.

Flashcard 40: Explain why partial fractions cannot be used if the fraction is improper.

Answer: Numerator degree must be less than denominator. Improper fractions need polynomial division first.

Flashcard 41: Identify the partial fractions for $\frac{x^2+3x+2}{x^3-x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$ . Factor $x^3-x = x(x-1)(x+1)$ .

Flashcard 42: Find the partial fractions for $\frac{x+5}{x^2+5x+6}$ .

Answer: $\frac{A}{x+2} + \frac{B}{x+3}$ . Factor $x^2+5x+6 = (x+2)(x+3)$ .

Flashcard 43: What is the general form of a linear partial fraction decomposition?

Answer: $\frac{A}{x-a} + \frac{B}{x-b}$ . For distinct linear factors, each gets one term.

Flashcard 44: What is the form of partial fractions for a factor like $(x-a)^n$ ?

Answer: $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$ . Each power of repeated factor needs its own term.

Flashcard 45: Identify the partial decomposition for $\frac{3x+1}{(x-3)^2}$ .

Answer: $\frac{A}{x-3} + \frac{B}{(x-3)^2}$ . Repeated factor requires two terms.

Flashcard 46: State the partial fraction form for $\frac{6x+9}{x^3(x-1)}$ .

Answer: $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}$ . $x^3$ factor creates three terms plus linear factor.

Flashcard 47: What is the purpose of partial fraction decomposition in calculus?

Answer: To simplify integration or differentiation. Converts complex fractions into simpler integrable forms.

Flashcard 48: Determine $A$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}$ .

Answer: $A = 3$ . Set $x=0$ to isolate $A$ .

Flashcard 49: Identify the partial fractions for $\frac{5x}{x^2+3x}$ .

Answer: $\frac{A}{x} + \frac{B}{x+3}$ . Factor out $x$ from denominator first.

Flashcard 50: What must be done if the rational function is improper for partial fractions?

Answer: Perform polynomial long division first. Convert improper to proper fraction first.

Flashcard 51: What is the advantage of using partial fractions in integration?

Answer: Simplifies the integral into manageable parts. Each simple fraction integrates easily.

Flashcard 52: What is the purpose of equating coefficients in partial fraction decomposition?

Answer: To solve for unknown constants. Matching coefficients determines constants.

Flashcard 53: Find the partial fractions for $\frac{x+6}{x^2-5x+6}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x-3}$ . Factor $x^2-5x+6 = (x-2)(x-3)$ .

Flashcard 54: Determine the partial fractions for $\frac{7x+4}{x^2-1}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x+1}$ . Factor $x^2-1 = (x-1)(x+1)$ .

Flashcard 55: Identify the partial fractions for $\frac{7x+9}{x^2+x}$ .

Answer: $\frac{A}{x} + \frac{B}{x+1}$ . Factor $x^2+x = x(x+1)$ .

Flashcard 56: Explain why partial fractions cannot be used if the fraction is improper.

Answer: Numerator degree must be less than denominator. Improper fractions need polynomial division first.

Flashcard 57: Identify the partial fraction decomposition for $\frac{1}{x(x+1)}$ .

Answer: $\frac{A}{x} + \frac{B}{x+1}$ . Factor denominator $x(x+1)$ , then assign constants.

Flashcard 58: Identify the partial fractions for $\frac{2x+1}{x^2+x-6}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x+3}$ . Factor $x^2+x-6 = (x-2)(x+3)$ .

Flashcard 59: Identify the partial fractions for $\frac{x^2+3x+2}{x^3-x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$ . Factor $x^3-x = x(x-1)(x+1)$ .

Flashcard 60: Find the partial fraction decomposition of $\frac{2x+3}{(x-1)(x+2)}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x+2}$ . Factor $(x-1)(x+2)$ , assign constants to each.

Flashcard 61: What is the form of partial fractions for a repeated linear factor like $(x-a)^2$ ?

Answer: $\frac{A}{x-a} + \frac{B}{(x-a)^2}$ . Repeated factors need terms for each power.

Flashcard 62: What is the denominator structure required for partial fraction decomposition?

Answer: Product of linear or irreducible quadratic factors. Standard form required for partial fraction method.

Flashcard 63: What is the benefit of using partial fractions in solving integrals?

Answer: Facilitates integration by breaking into simpler terms. Each partial fraction integrates using basic rules.

Flashcard 64: What is a key benefit of partial fraction decomposition for solving integrals?

Answer: Breaks down complex rational functions. Makes integration much simpler.

Flashcard 65: Find the partial fraction decomposition of $\frac{2x+3}{(x-1)(x+2)}$ .

Answer: $\frac{A}{x-1} + \frac{B}{x+2}$ . Factor $(x-1)(x+2)$ , assign constants to each.

Flashcard 66: What substitution is used to solve for $A$ in partial fractions?

Answer: Set $x$ to make other term zero. Strategic substitution eliminates other terms.

Flashcard 67: Determine the partial fractions for $\frac{3x^2}{x^3-x}$ .

Answer: $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$ . Factor $x^3-x = x(x-1)(x+1)$ .

Flashcard 68: Find the partial fractions for $\frac{x+5}{x^2+5x+6}$ .

Answer: $\frac{A}{x+2} + \frac{B}{x+3}$ . Factor $x^2+5x+6 = (x+2)(x+3)$ .

Flashcard 69: Identify the partial fractions for $\frac{5}{x^2-4x+4}$ .

Answer: $\frac{A}{x-2} + \frac{B}{(x-2)^2}$ . Factor $x^2-4x+4 = (x-2)^2$ .

Flashcard 70: Identify the partial fractions for $\frac{5}{x^2-4x+4}$ .

Answer: $\frac{A}{x-2} + \frac{B}{(x-2)^2}$ . Factor $x^2-4x+4 = (x-2)^2$ .

Flashcard 71: Find the partial fraction decomposition of $\frac{4x+2}{x^2-4}$ .

Answer: $\frac{A}{x-2} + \frac{B}{x+2}$ . Factor $x^2-4 = (x-2)(x+2)$ .

Flashcard 72: What must be true about the degree of the numerator relative to the denominator?

Answer: Degree of numerator < degree of denominator. Proper fractions are required for decomposition.

Flashcard 73: Find $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}$ .

Answer: $B = 5$ . Set $x=-2$ to isolate $B$ .

Flashcard 74: What is the benefit of using partial fractions in solving integrals?

Answer: Facilitates integration by breaking into simpler terms. Each partial fraction integrates using basic rules.

AP Calculus BC Flashcards: Using Linear Partial Fractions

Using Linear Partial Fractions

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Calculate BBB in the decomposition Ax+Bx+1=4x(x+1)\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}xA​+x+1B​=x(x+1)4​.

Flashcard 2: What is the result of using partial fractions on 1(x+1)2\frac{1}{(x+1)^2}(x+1)21​?

Flashcard 3: Determine the partial fractions for 8xx2−2x\frac{8x}{x^2-2x}x2−2x8x​.

Flashcard 4: Determine the partial fractions for 8xx2−2x\frac{8x}{x^2-2x}x2−2x8x​.

Flashcard 5: What is the general form of a linear partial fraction decomposition?

Flashcard 6: What is the first step in decomposing a rational function into partial fractions?

Flashcard 7: What is the first step in decomposing a rational function into partial fractions?

Flashcard 8: Identify the partial fractions for 7x+9x2+x\frac{7x+9}{x^2+x}x2+x7x+9​.

Flashcard 9: Find AAA in the decomposition Ax+Bx−1=2x+3x(x−1)\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}xA​+x−1B​=x(x−1)2x+3​.

Flashcard 10: What is the denominator structure required for partial fraction decomposition?

Flashcard 11: What is the purpose of equating coefficients in partial fraction decomposition?

Flashcard 12: What is the advantage of using partial fractions in integration?

Flashcard 13: Determine the partial fractions for 7x+4x2−1\frac{7x+4}{x^2-1}x2−17x+4​.

Flashcard 14: What does each term in a partial fraction decomposition represent?

Flashcard 15: Find AAA in the decomposition Ax+Bx−1=2x+3x(x−1)\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}xA​+x−1B​=x(x−1)2x+3​.

Flashcard 16: What is the partial fraction form for a simple linear factor like x−ax-ax−a?

Flashcard 17: Find BBB in the decomposition Ax+Bx+2=2x+5x(x+2)\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}xA​+x+2B​=x(x+2)2x+5​.

Flashcard 18: Determine AAA in the decomposition Ax+Bx+1=3x(x+1)\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}xA​+x+1B​=x(x+1)3​.

Flashcard 19: What does each term in a partial fraction decomposition represent?

Flashcard 20: Calculate BBB in the decomposition Ax+Bx+1=4x(x+1)\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}xA​+x+1B​=x(x+1)4​.

Flashcard 21: Determine the partial fractions for 3x2x3−x\frac{3x^2}{x^3-x}x3−x3x2​.

Flashcard 22: What is the result of using partial fractions on 1(x+1)2\frac{1}{(x+1)^2}(x+1)21​?

Flashcard 23: Identify the partial fraction decomposition for 1x(x+1)\frac{1}{x(x+1)}x(x+1)1​.

Flashcard 24: What must be true about the degree of the numerator relative to the denominator?

Flashcard 25: Find the partial fractions for x+6x2−5x+6\frac{x+6}{x^2-5x+6}x2−5x+6x+6​.

Flashcard 26: Identify the partial fraction form for 1(x−1)(x−2)(x−3)\frac{1}{(x-1)(x-2)(x-3)}(x−1)(x−2)(x−3)1​.

Flashcard 27: Find the partial fraction decomposition of 4x+2x2−4\frac{4x+2}{x^2-4}x2−44x+2​.

Flashcard 28: What is the form of partial fractions for a factor like (x−a)n(x-a)^n(x−a)n?

Flashcard 29: What substitution is used to solve for AAA in partial fractions?

Flashcard 30: Identify the partial fraction form for 1(x−1)(x−2)(x−3)\frac{1}{(x-1)(x-2)(x-3)}(x−1)(x−2)(x−3)1​.

Flashcard 31: What is the purpose of partial fraction decomposition in calculus?

Flashcard 32: Identify the partial fractions for 2x+1x2+x−6\frac{2x+1}{x^2+x-6}x2+x−62x+1​.

Flashcard 33: Identify the partial decomposition for 3x+1(x−3)2\frac{3x+1}{(x-3)^2}(x−3)23x+1​.

Flashcard 34: Identify the partial fractions for 5xx2+3x\frac{5x}{x^2+3x}x2+3x5x​.

Flashcard 35: What is the form of partial fractions for a repeated linear factor like (x−a)2(x-a)^2(x−a)2?

Flashcard 36: State the partial fraction form for 6x+9x3(x−1)\frac{6x+9}{x^3(x-1)}x3(x−1)6x+9​.

Flashcard 37: What is the partial fraction form for a simple linear factor like x−ax-ax−a?

Flashcard 38: What must be done if the rational function is improper for partial fractions?

Flashcard 39: What is a key benefit of partial fraction decomposition for solving integrals?

Flashcard 40: Explain why partial fractions cannot be used if the fraction is improper.

Flashcard 41: Identify the partial fractions for x2+3x+2x3−x\frac{x^2+3x+2}{x^3-x}x3−xx2+3x+2​.

Flashcard 42: Find the partial fractions for x+5x2+5x+6\frac{x+5}{x^2+5x+6}x2+5x+6x+5​.

Flashcard 43: What is the general form of a linear partial fraction decomposition?

Flashcard 44: What is the form of partial fractions for a factor like (x−a)n(x-a)^n(x−a)n?

Flashcard 45: Identify the partial decomposition for 3x+1(x−3)2\frac{3x+1}{(x-3)^2}(x−3)23x+1​.

Flashcard 46: State the partial fraction form for 6x+9x3(x−1)\frac{6x+9}{x^3(x-1)}x3(x−1)6x+9​.

Flashcard 47: What is the purpose of partial fraction decomposition in calculus?

Flashcard 48: Determine AAA in the decomposition Ax+Bx+1=3x(x+1)\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}xA​+x+1B​=x(x+1)3​.

Flashcard 49: Identify the partial fractions for 5xx2+3x\frac{5x}{x^2+3x}x2+3x5x​.

Flashcard 50: What must be done if the rational function is improper for partial fractions?

Flashcard 51: What is the advantage of using partial fractions in integration?

Flashcard 52: What is the purpose of equating coefficients in partial fraction decomposition?

Flashcard 53: Find the partial fractions for x+6x2−5x+6\frac{x+6}{x^2-5x+6}x2−5x+6x+6​.

Flashcard 54: Determine the partial fractions for 7x+4x2−1\frac{7x+4}{x^2-1}x2−17x+4​.

Flashcard 55: Identify the partial fractions for 7x+9x2+x\frac{7x+9}{x^2+x}x2+x7x+9​.

Flashcard 56: Explain why partial fractions cannot be used if the fraction is improper.

Flashcard 57: Identify the partial fraction decomposition for 1x(x+1)\frac{1}{x(x+1)}x(x+1)1​.

Flashcard 58: Identify the partial fractions for 2x+1x2+x−6\frac{2x+1}{x^2+x-6}x2+x−62x+1​.

Flashcard 59: Identify the partial fractions for x2+3x+2x3−x\frac{x^2+3x+2}{x^3-x}x3−xx2+3x+2​.

Flashcard 60: Find the partial fraction decomposition of 2x+3(x−1)(x+2)\frac{2x+3}{(x-1)(x+2)}(x−1)(x+2)2x+3​.

Flashcard 61: What is the form of partial fractions for a repeated linear factor like (x−a)2(x-a)^2(x−a)2?

Flashcard 62: What is the denominator structure required for partial fraction decomposition?

Flashcard 63: What is the benefit of using partial fractions in solving integrals?

Flashcard 64: What is a key benefit of partial fraction decomposition for solving integrals?

Flashcard 65: Find the partial fraction decomposition of 2x+3(x−1)(x+2)\frac{2x+3}{(x-1)(x+2)}(x−1)(x+2)2x+3​.

Flashcard 66: What substitution is used to solve for AAA in partial fractions?

Flashcard 67: Determine the partial fractions for 3x2x3−x\frac{3x^2}{x^3-x}x3−x3x2​.

Flashcard 68: Find the partial fractions for x+5x2+5x+6\frac{x+5}{x^2+5x+6}x2+5x+6x+5​.

Flashcard 69: Identify the partial fractions for 5x2−4x+4\frac{5}{x^2-4x+4}x2−4x+45​.

Flashcard 70: Identify the partial fractions for 5x2−4x+4\frac{5}{x^2-4x+4}x2−4x+45​.

Flashcard 71: Find the partial fraction decomposition of 4x+2x2−4\frac{4x+2}{x^2-4}x2−44x+2​.

Flashcard 72: What must be true about the degree of the numerator relative to the denominator?

Flashcard 73: Find BBB in the decomposition Ax+Bx+2=2x+5x(x+2)\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}xA​+x+2B​=x(x+2)2x+5​.

Flashcard 74: What is the benefit of using partial fractions in solving integrals?

Flashcard 1: Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$ .

Flashcard 2: What is the result of using partial fractions on $\frac{1}{(x+1)^2}$ ?

Flashcard 3: Determine the partial fractions for $\frac{8x}{x^2-2x}$ .

Flashcard 4: Determine the partial fractions for $\frac{8x}{x^2-2x}$ .

Flashcard 8: Identify the partial fractions for $\frac{7x+9}{x^2+x}$ .

Flashcard 9: Find $A$ in the decomposition $\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}$ .

Flashcard 13: Determine the partial fractions for $\frac{7x+4}{x^2-1}$ .

Flashcard 15: Find $A$ in the decomposition $\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}$ .

Flashcard 16: What is the partial fraction form for a simple linear factor like $x-a$ ?

Flashcard 17: Find $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}$ .

Flashcard 18: Determine $A$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}$ .

Flashcard 20: Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$ .

Flashcard 21: Determine the partial fractions for $\frac{3x^2}{x^3-x}$ .

Flashcard 22: What is the result of using partial fractions on $\frac{1}{(x+1)^2}$ ?

Flashcard 23: Identify the partial fraction decomposition for $\frac{1}{x(x+1)}$ .

Flashcard 25: Find the partial fractions for $\frac{x+6}{x^2-5x+6}$ .

Flashcard 26: Identify the partial fraction form for $\frac{1}{(x-1)(x-2)(x-3)}$ .

Flashcard 27: Find the partial fraction decomposition of $\frac{4x+2}{x^2-4}$ .

Flashcard 28: What is the form of partial fractions for a factor like $(x-a)^n$ ?

Flashcard 29: What substitution is used to solve for $A$ in partial fractions?

Flashcard 30: Identify the partial fraction form for $\frac{1}{(x-1)(x-2)(x-3)}$ .

Flashcard 32: Identify the partial fractions for $\frac{2x+1}{x^2+x-6}$ .

Flashcard 33: Identify the partial decomposition for $\frac{3x+1}{(x-3)^2}$ .

Flashcard 34: Identify the partial fractions for $\frac{5x}{x^2+3x}$ .

Flashcard 35: What is the form of partial fractions for a repeated linear factor like $(x-a)^2$ ?

Flashcard 36: State the partial fraction form for $\frac{6x+9}{x^3(x-1)}$ .

Flashcard 37: What is the partial fraction form for a simple linear factor like $x-a$ ?

Flashcard 41: Identify the partial fractions for $\frac{x^2+3x+2}{x^3-x}$ .

Flashcard 42: Find the partial fractions for $\frac{x+5}{x^2+5x+6}$ .

Flashcard 44: What is the form of partial fractions for a factor like $(x-a)^n$ ?

Flashcard 45: Identify the partial decomposition for $\frac{3x+1}{(x-3)^2}$ .

Flashcard 46: State the partial fraction form for $\frac{6x+9}{x^3(x-1)}$ .

Flashcard 48: Determine $A$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}$ .

Flashcard 49: Identify the partial fractions for $\frac{5x}{x^2+3x}$ .

Flashcard 53: Find the partial fractions for $\frac{x+6}{x^2-5x+6}$ .

Flashcard 54: Determine the partial fractions for $\frac{7x+4}{x^2-1}$ .

Flashcard 55: Identify the partial fractions for $\frac{7x+9}{x^2+x}$ .

Flashcard 57: Identify the partial fraction decomposition for $\frac{1}{x(x+1)}$ .

Flashcard 58: Identify the partial fractions for $\frac{2x+1}{x^2+x-6}$ .

Flashcard 59: Identify the partial fractions for $\frac{x^2+3x+2}{x^3-x}$ .

Flashcard 60: Find the partial fraction decomposition of $\frac{2x+3}{(x-1)(x+2)}$ .

Flashcard 61: What is the form of partial fractions for a repeated linear factor like $(x-a)^2$ ?

Flashcard 65: Find the partial fraction decomposition of $\frac{2x+3}{(x-1)(x+2)}$ .

Flashcard 66: What substitution is used to solve for $A$ in partial fractions?

Flashcard 67: Determine the partial fractions for $\frac{3x^2}{x^3-x}$ .

Flashcard 68: Find the partial fractions for $\frac{x+5}{x^2+5x+6}$ .

Flashcard 69: Identify the partial fractions for $\frac{5}{x^2-4x+4}$ .

Flashcard 70: Identify the partial fractions for $\frac{5}{x^2-4x+4}$ .

Flashcard 71: Find the partial fraction decomposition of $\frac{4x+2}{x^2-4}$ .

Flashcard 73: Find $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}$ .