AP Calculus AB Flashcards: Integrating Long Division Completing The Square

Study Integrating Long Division Completing The Square in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Integrating Long Division Completing The Square

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QUESTION

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Identify the completed square form of $x^2 + 6x + 11$ .

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ANSWER

$(x + 3)^2 + 2$ . Take half the coefficient of $x$ , square it: $(\frac{6}{2})^2 = 9$ , so add/subtract 9.

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What this deck covers

This deck focuses on Integrating Long Division Completing The Square, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

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: Identify the completed square form of $x^2 + 6x + 11$ .

Answer: $(x + 3)^2 + 2$ . Take half the coefficient of $x$ , square it: $(\frac{6}{2})^2 = 9$ , so add/subtract 9.

Flashcard 2: What is the completed square form of $x^2 + 2x + 3$ ?

Answer: $(x+1)^2 + 2$ . Complete the square: $(x+1)^2 + 2$ where $2 = 3 - 1$ .

Flashcard 3: Find the completed square form of $x^2 + 2x + 1$ .

Answer: $(x+1)^2$ . Perfect square trinomial: $(x+1)^2$ .

Flashcard 4: What is the purpose of completing the square in integration?

Answer: To simplify the integrand into a recognizable form. Transforms expressions into standard forms that match integration formulas.

Flashcard 5: Convert $x^2 + 4x + 5$ into completed square form.

Answer: $(x+2)^2 + 1$ . Complete the square: $x^2 + 4x + 4 + 1 = (x+2)^2 + 1$ .

Flashcard 6: What is the integral of $\frac{1}{(x+2)^2 + 2}$ ?

Answer: $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+2}{\sqrt{2}} + C$ . Arctangent formula with $a = \sqrt{2}$ .

Flashcard 7: Use long division for $\frac{x^3 + 6x^2 + 11x + 6}{x + 3}$ . What is the quotient?

Answer: $x^2 + 3x + 2$ . Divide cubic by linear using polynomial long division.

Flashcard 8: Express $x^2 + 4x + 13$ in completed square form.

Answer: $(x+2)^2 + 9$ . Complete the square by adding and subtracting $(\frac{4}{2})^2 = 4$ .

Flashcard 9: Express $x^2 + 14x + 49$ in completed square form.

Answer: $(x+7)^2$ . This is a perfect square trinomial: $(x+7)^2$ .

Flashcard 10: Express $x^2 + 2x + 2$ in completed square form.

Answer: $(x+1)^2 + 1$ . Add and subtract $(\frac{2}{2})^2 = 1$ to complete the square.

Flashcard 11: What is the first step when integrating $\frac{x^3 + 2x^2 + x + 1}{x^2 + 1}$ using long division?

Answer: Divide $x^3 + 2x^2 + x + 1$ by $x^2 + 1$ . Use polynomial long division when the degree of numerator ≥ degree of denominator.

Flashcard 12: Find the completed square form of $x^2 + 10x + 25$ .

Answer: $(x+5)^2$ . This is already a perfect square trinomial.

Flashcard 13: State the integral of $\frac{1}{x^2 + 2x + 3}$ after completing the square.

Answer: Use $\frac{1}{(x+1)^2 + 2}$ form. After completing the square: $x^2 + 2x + 3 = (x+1)^2 + 2$ .

Flashcard 14: Express $x^2 + 14x + 49$ in completed square form.

Answer: $(x+7)^2$ . This is a perfect square trinomial: $(x+7)^2$ .

Flashcard 15: What is the integral of $\frac{1}{(x+3)^2 + 1}$ ?

Answer: $\tan^{-1}(x+3) + C$ . Standard arctangent integral form.

Flashcard 16: What is the integral of $\frac{1}{(x+3)^2 + 1}$ ?

Answer: $\tan^{-1}(x+3) + C$ . Standard arctangent integral form.

Flashcard 17: Convert $x^2 + 12x + 36$ into a completed square.

Answer: $(x+6)^2$ . Perfect square trinomial with $a = 6$ .

Flashcard 18: Use long division for $\frac{x^2 + 3x + 2}{x + 1}$ . What is the quotient?

Answer: $x + 2$ . Long division gives quotient and remainder.

Flashcard 19: What is the integral of $\frac{1}{(x+1)^2 + 4}$ ?

Answer: $\frac{1}{2} \tan^{-1}\frac{x+1}{2} + C$ . Arctangent formula with $a = 2$ .

Flashcard 20: What is the integral of $\frac{1}{(x+6)^2}$ ?

Answer: $-\frac{1}{x+6} + C$ . Power rule: $\int (x+6)^{-2} dx = -(x+6)^{-1} + C$ .

Flashcard 21: State the integral of $\frac{1}{x^2 + 2x + 3}$ after completing the square.

Answer: Use $\frac{1}{(x+1)^2 + 2}$ form. After completing the square: $x^2 + 2x + 3 = (x+1)^2 + 2$ .

Flashcard 22: What is the integral of $\frac{1}{(x+2)^2+1}$ ?

Answer: $\tan^{-1}(x+2) + C$ . Standard arctangent integral with $a = 1$ .

Flashcard 23: How do you handle $\frac{x^2 + 3x + 5}{x + 2}$ using long division?

Answer: Divide $x^2 + 3x + 5$ by $x + 2$ . Degree of numerator > degree of denominator requires polynomial division first.

Flashcard 24: What is the antiderivative of $\frac{1}{(x+1)^2}$ ?

Answer: $-\frac{1}{x+1} + C$ . Standard integral of $u^{-2}$ form.

Flashcard 25: Identify the integral of $\frac{1}{x^2 + 4x + 13}$ after completing the square.

Answer: Use $\frac{1}{(x+2)^2 + 9}$ form. Complete the square: $x^2 + 4x + 13 = (x+2)^2 + 9$ .

Flashcard 26: What is the integral of $\frac{1}{(x+3)^2 + 2}$ ?

Answer: $\frac{1}{\text{sqrt}(2)} \tan^{-1}\frac{x+3}{\text{sqrt}(2)} + C$ . Use the formula $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$ .

Flashcard 27: Convert $x^2 + 4x + 6$ into completed square form.

Answer: $(x+2)^2 + 2$ . Complete the square: $(x+2)^2 + 2$ where $2 = 6 - 4$ .

Flashcard 28: Convert $x^2 + 4x + 5$ into completed square form.

Answer: $(x+2)^2 + 1$ . Complete the square: $x^2 + 4x + 4 + 1 = (x+2)^2 + 1$ .

Flashcard 29: Convert $x^2 + 8x + 16$ into a completed square.

Answer: $(x+4)^2$ . This is already a perfect square trinomial.

Flashcard 30: What is the integral of $\frac{1}{(x+1)^2 + 2}$ ?

Answer: $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+1}{\sqrt{2}} + C$ . Use arctangent formula with $a = \sqrt{2}$ .

Flashcard 31: Find the completed square form of $x^2 + 10x + 25$ .

Answer: $(x+5)^2$ . This is already a perfect square trinomial.

Flashcard 32: What is the integral of $\frac{1}{(x+7)^2}$ ?

Answer: $-\frac{1}{x+7} + C$ . Standard power rule for $u^{-2}$

Flashcard 33: Express $x^2 + 2x + 2$ in completed square form.

Answer: $(x+1)^2 + 1$ . Add and subtract $(\frac{2}{2})^2 = 1$ to complete the square.

Flashcard 34: What is the completed square form of $x^2 + 2x + 3$ ?

Answer: $(x+1)^2 + 2$ . Complete the square: $(x+1)^2 + 2$ where $2 = 3 - 1$ .

Flashcard 35: What is the integral of $\frac{1}{(x+2)^2 + 2}$ ?

Answer: $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+2}{\sqrt{2}} + C$ . Arctangent formula with $a = \sqrt{2}$ .

Flashcard 36: What is the integral of $\frac{1}{(x+3)^2 + 2}$ ?

Answer: $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+3}{\sqrt{2}} + C$ . Use the formula $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$ .

Flashcard 37: What is the integral of $\frac{1}{(x+2)^2 + 4}$ ?

Answer: $\frac{1}{2} \tan^{-1}\frac{x+2}{2} + C$ . Use $\int \frac{1}{x^2 + a^2} dx$ formula with $a = 2$ .

Flashcard 38: Use long division for $\frac{x^2 + 3x + 2}{x + 1}$ . What is the quotient?

Answer: $x + 2$ . Long division gives quotient and remainder.

Flashcard 39: Perform long division on $\frac{x^2 + 5x + 6}{x + 1}$ . What is the quotient?

Answer: $x + 4$ . Polynomial long division of quadratic by linear.

Flashcard 40: How do you handle $\frac{x^2 + 3x + 5}{x + 2}$ using long division?

Answer: Divide $x^2 + 3x + 5$ by $x + 2$ . Degree of numerator > degree of denominator requires polynomial division first.

Flashcard 41: What is the integral of $\frac{1}{(x+1)^2 + 2}$ ?

Answer: $\frac{1}{\text{sqrt}(2)} \tan^{-1}\frac{x+1}{\text{sqrt}(2)} + C$ . Use arctangent formula with $a = \sqrt{2}$ .

Flashcard 42: Use long division for $\frac{x^3 + 6x^2 + 11x + 6}{x + 3}$ . What is the quotient?

Answer: $x^2 + 3x + 2$ . Divide cubic by linear using polynomial long division.

Flashcard 43: What is the integral of $\frac{1}{(x+5)^2}$ ?

Answer: $-\frac{1}{x+5} + C$ . Integral of $u^{-2}$ is $-u^{-1} + C$ .

Flashcard 44: What is the first step in completing the square for $x^2 + 10x + 29$ ?

Answer: Write as $(x+5)^2 + 4$ . Take half the coefficient of $x$ : $(\frac{10}{2})^2 = 25$ , then adjust.

Flashcard 45: Convert $x^2 + 12x + 36$ into a completed square.

Answer: $(x+6)^2$ . Perfect square trinomial with $a = 6$ .

Flashcard 46: What is the integral of $\frac{1}{(x+4)^2}$ ?

Answer: $-\frac{1}{x+4} + C$ . Use the power rule: $\int u^{-2} du = -u^{-1} + C$ .

Flashcard 47: What is the first step when integrating $\frac{x^3 + 2x^2 + x + 1}{x^2 + 1}$ using long division?

Answer: Divide $x^3 + 2x^2 + x + 1$ by $x^2 + 1$ . Use polynomial long division when the degree of numerator ≥ degree of denominator.

Flashcard 48: What is the purpose of completing the square in integration?

Answer: To simplify the integrand into a recognizable form. Transforms expressions into standard forms that match integration formulas.

Flashcard 49: What is the integral of $\frac{1}{(x+6)^2}$ ?

Answer: $-\frac{1}{x+6} + C$ . Power rule: $\int (x+6)^{-2} dx = -(x+6)^{-1} + C$ .

Flashcard 50: Find the completed square form of $x^2 + 2x + 1$ .

Answer: $(x+1)^2$ . Perfect square trinomial: $(x+1)^2$ .

Flashcard 51: What is the integral of $\frac{1}{(x+2)^2+1}$ ?

Answer: $\tan^{-1}(x+2) + C$ . Standard arctangent integral with $a = 1$ .

Flashcard 52: Express $x^2 + 6x + 10$ in completed square form.

Answer: $(x+3)^2 + 1$ . Complete the square: $(x+3)^2 + 1$ where $1 = 10 - 9$ .

Flashcard 53: Convert $x^2 + 2x + 5$ into completed square form.

Answer: $(x+1)^2 + 4$ . Complete the square: add/subtract $(\frac{2}{2})^2 = 1$

Flashcard 54: What is the antiderivative of $\frac{1}{(x+2)^2 + 9}$ ?

Answer: $\frac{1}{3} \tan^{-1}\left(\frac{x+2}{3}\right) + C$ . Use arctangent formula with $a = 3$ : $\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$ .

Flashcard 55: Convert $x^2 + 8x + 16$ into a completed square.

Answer: $(x+4)^2$ . This is already a perfect square trinomial.

Flashcard 56: Convert $x^2 + 2x + 5$ into completed square form.

Answer: $(x+1)^2 + 4$ . Complete the square: add/subtract $(\frac{2}{2})^2 = 1$ .

Flashcard 57: What is the integral of $\frac{1}{(x+1)^2 + 4}$ ?

Answer: $\frac{1}{2} \tan^{-1}\frac{x+1}{2} + C$ . Arctangent formula with $a = 2$ .

Flashcard 58: Identify the integral of $\frac{1}{x^2 + 4x + 13}$ after completing the square.

Answer: Use $\frac{1}{(x+2)^2 + 9}$ form. Complete the square: $x^2 + 4x + 13 = (x+2)^2 + 9$ .

Flashcard 59: What is the antiderivative of $\frac{1}{(x+2)^2 + 9}$ ?

Answer: $\frac{1}{3} \tan^{-1} \left( \frac{x+2}{3} \right) + C$ . Use arctangent formula with $a = 3$ : $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$ .

Flashcard 60: What is the integral of $\frac{1}{(x+1)^2 + 1}$ ?

Answer: $\tan^{-1}(x+1) + C$ . Arctangent integral with $a = 1$ .

Flashcard 61: Perform long division on $\frac{x^2 + x + 1}{x + 1}$ . What is the quotient?

Answer: $x$ . Polynomial division of $x^2 + x + 1$ by $x + 1$ .

Flashcard 62: What is the integral of $\frac{1}{(x+1)^2 + 1}$ ?

Answer: $\tan^{-1}(x+1) + C$ . Arctangent integral with $a = 1$ .

Flashcard 63: What is the integral of $\frac{1}{(x+2)^2 + 4}$ ?

Answer: $\frac{1}{2} \tan^{-1}\frac{x+2}{2} + C$ . Use $\int \frac{1}{x^2 + a^2} dx$ formula with $a = 2$ .

Flashcard 64: What is the completed square form of $x^2 + 4x + 8$ ?

Answer: $(x+2)^2 + 4$ . Complete the square: $(x+2)^2 + 4$ where $4 = 8 - 4$ .

Flashcard 65: Perform long division on $\frac{x^2 + 5x + 6}{x + 1}$ . What is the quotient?

Answer: $x + 4$ . Polynomial long division of quadratic by linear.

Flashcard 66: What is the first step in completing the square for $x^2 + 10x + 29$ ?

Answer: Write as $(x+5)^2 + 4$ . Take half the coefficient of $x$ : $(\frac{10}{2})^2 = 25$ , then adjust.

Flashcard 67: What is the integral of $\frac{1}{(x+5)^2}$ ?

Answer: $-\frac{1}{x+5} + C$ . Integral of $u^{-2}$ is $-u^{-1} + C$ .

Flashcard 68: Identify the completed square form of $x^2 + 6x + 11$ .

Answer: $(x + 3)^2 + 2$ . Take half the coefficient of $x$ , square it: $(\frac{6}{2})^2 = 9$ , so add/subtract 9.

Flashcard 69: Convert $x^2 + 4x + 6$ into completed square form.

Answer: $(x+2)^2 + 2$ . Complete the square: $(x+2)^2 + 2$ where $2 = 6 - 4$ .

Flashcard 70: What is the antiderivative of $\frac{1}{(x+1)^2}$ ?

Answer: $-\frac{1}{x+1} + C$ . Standard integral of $u^{-2}$ form.

Flashcard 71: Express $x^2 + 6x + 10$ in completed square form.

Answer: $(x+3)^2 + 1$ . Complete the square: $(x+3)^2 + 1$ where $1 = 10 - 9$ .

Flashcard 72: Express $x^2 + 4x + 13$ in completed square form.

Answer: $(x+2)^2 + 9$ . Complete the square by adding and subtracting $(\frac{4}{2})^2 = 4$ .

Flashcard 73: What is the completed square form of $x^2 + 4x + 8$ ?

Answer: $(x+2)^2 + 4$ . Complete the square: $(x+2)^2 + 4$ where $4 = 8 - 4$ .

Flashcard 74: Perform long division on $\frac{x^2 + x + 1}{x + 1}$ . What is the quotient?

Answer: $x$ . Polynomial division of $x^2 + x + 1$ by $x + 1$ .

Flashcard 75: What is the integral of $\frac{1}{(x+4)^2}$ ?

Answer: $-\frac{1}{x+4} + C$ . Use the power rule: $\int u^{-2} du = -u^{-1} + C$ .

AP Calculus AB Flashcards: Integrating Long Division Completing The Square

Integrating Long Division Completing The Square

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Identify the completed square form of x2+6x+11x^2 + 6x + 11x2+6x+11.

Flashcard 2: What is the completed square form of x2+2x+3x^2 + 2x + 3x2+2x+3?

Flashcard 3: Find the completed square form of x2+2x+1x^2 + 2x + 1x2+2x+1.

Flashcard 4: What is the purpose of completing the square in integration?

Flashcard 5: Convert x2+4x+5x^2 + 4x + 5x2+4x+5 into completed square form.

Flashcard 6: What is the integral of 1(x+2)2+2\frac{1}{(x+2)^2 + 2}(x+2)2+21​?

Flashcard 7: Use long division for x3+6x2+11x+6x+3\frac{x^3 + 6x^2 + 11x + 6}{x + 3}x+3x3+6x2+11x+6​. What is the quotient?

Flashcard 8: Express x2+4x+13x^2 + 4x + 13x2+4x+13 in completed square form.

Flashcard 9: Express x2+14x+49x^2 + 14x + 49x2+14x+49 in completed square form.

Flashcard 10: Express x2+2x+2x^2 + 2x + 2x2+2x+2 in completed square form.

Flashcard 11: What is the first step when integrating x3+2x2+x+1x2+1\frac{x^3 + 2x^2 + x + 1}{x^2 + 1}x2+1x3+2x2+x+1​ using long division?

Flashcard 12: Find the completed square form of x2+10x+25x^2 + 10x + 25x2+10x+25.

Flashcard 13: State the integral of 1x2+2x+3\frac{1}{x^2 + 2x + 3}x2+2x+31​ after completing the square.

Flashcard 14: Express x2+14x+49x^2 + 14x + 49x2+14x+49 in completed square form.

Flashcard 15: What is the integral of 1(x+3)2+1\frac{1}{(x+3)^2 + 1}(x+3)2+11​?

Flashcard 16: What is the integral of 1(x+3)2+1\frac{1}{(x+3)^2 + 1}(x+3)2+11​?

Flashcard 17: Convert x2+12x+36x^2 + 12x + 36x2+12x+36 into a completed square.

Flashcard 18: Use long division for x2+3x+2x+1\frac{x^2 + 3x + 2}{x + 1}x+1x2+3x+2​. What is the quotient?

Flashcard 19: What is the integral of 1(x+1)2+4\frac{1}{(x+1)^2 + 4}(x+1)2+41​?

Flashcard 20: What is the integral of 1(x+6)2\frac{1}{(x+6)^2}(x+6)21​?

Flashcard 21: State the integral of 1x2+2x+3\frac{1}{x^2 + 2x + 3}x2+2x+31​ after completing the square.

Flashcard 22: What is the integral of 1(x+2)2+1\frac{1}{(x+2)^2+1}(x+2)2+11​?

Flashcard 23: How do you handle x2+3x+5x+2\frac{x^2 + 3x + 5}{x + 2}x+2x2+3x+5​ using long division?

Flashcard 24: What is the antiderivative of 1(x+1)2\frac{1}{(x+1)^2}(x+1)21​?

Flashcard 25: Identify the integral of 1x2+4x+13\frac{1}{x^2 + 4x + 13}x2+4x+131​ after completing the square.

Flashcard 26: What is the integral of 1(x+3)2+2\frac{1}{(x+3)^2 + 2}(x+3)2+21​?

Flashcard 27: Convert x2+4x+6x^2 + 4x + 6x2+4x+6 into completed square form.

Flashcard 28: Convert x2+4x+5x^2 + 4x + 5x2+4x+5 into completed square form.

Flashcard 29: Convert x2+8x+16x^2 + 8x + 16x2+8x+16 into a completed square.

Flashcard 30: What is the integral of 1(x+1)2+2\frac{1}{(x+1)^2 + 2}(x+1)2+21​?

Flashcard 31: Find the completed square form of x2+10x+25x^2 + 10x + 25x2+10x+25.

Flashcard 32: What is the integral of 1(x+7)2\frac{1}{(x+7)^2}(x+7)21​?

Flashcard 33: Express x2+2x+2x^2 + 2x + 2x2+2x+2 in completed square form.

Flashcard 34: What is the completed square form of x2+2x+3x^2 + 2x + 3x2+2x+3?

Flashcard 35: What is the integral of 1(x+2)2+2\frac{1}{(x+2)^2 + 2}(x+2)2+21​?

Flashcard 36: What is the integral of 1(x+3)2+2\frac{1}{(x+3)^2 + 2}(x+3)2+21​?

Flashcard 37: What is the integral of 1(x+2)2+4\frac{1}{(x+2)^2 + 4}(x+2)2+41​?

Flashcard 38: Use long division for x2+3x+2x+1\frac{x^2 + 3x + 2}{x + 1}x+1x2+3x+2​. What is the quotient?

Flashcard 39: Perform long division on x2+5x+6x+1\frac{x^2 + 5x + 6}{x + 1}x+1x2+5x+6​. What is the quotient?

Flashcard 40: How do you handle x2+3x+5x+2\frac{x^2 + 3x + 5}{x + 2}x+2x2+3x+5​ using long division?

Flashcard 41: What is the integral of 1(x+1)2+2\frac{1}{(x+1)^2 + 2}(x+1)2+21​?

Flashcard 42: Use long division for x3+6x2+11x+6x+3\frac{x^3 + 6x^2 + 11x + 6}{x + 3}x+3x3+6x2+11x+6​. What is the quotient?

Flashcard 43: What is the integral of 1(x+5)2\frac{1}{(x+5)^2}(x+5)21​?

Flashcard 44: What is the first step in completing the square for x2+10x+29x^2 + 10x + 29x2+10x+29?

Flashcard 45: Convert x2+12x+36x^2 + 12x + 36x2+12x+36 into a completed square.

Flashcard 46: What is the integral of 1(x+4)2\frac{1}{(x+4)^2}(x+4)21​?

Flashcard 47: What is the first step when integrating x3+2x2+x+1x2+1\frac{x^3 + 2x^2 + x + 1}{x^2 + 1}x2+1x3+2x2+x+1​ using long division?