AP Calculus BC Flashcards: Integrating Long Division Completing The Square

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 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.

Flashcard 1: Why is completing the square helpful in integration?

Answer: Transforms to a form suitable for standard integrals. Creates forms matching known integral formulas like $\frac{1}{u^2+a^2}$.

Flashcard 2: Integrate $\frac{1}{x^2 - 4}$. What method simplifies it?

Answer: Partial fraction decomposition. Factor as $(x-2)(x+2)$ for partial fraction decomposition.

Flashcard 3: What is the integral of $1/x$?

Answer: $\text{ln}|x| + C$. The antiderivative of the reciprocal function.

Flashcard 4: Perform polynomial division: $\frac{x^3 - 6x^2 + 11x - 6}{x - 2}$.

Answer: $x^2 - 4x + 3$. Synthetic or long division of cubic by linear factor.

Flashcard 5: What integral technique uses $u$ substitution on quadratics?

Answer: Completing the square first. Complete the square first, then substitute $u = x + h$.

Flashcard 6: What is the benefit of transforming a quadratic in integration?

Answer: Simplifies the integral into standard forms. Standard forms match known antiderivative formulas.

Flashcard 7: When dividing $x^3 - 1$ by $x - 1$, what do you get?

Answer: $x^2 + x + 1$. Factor theorem: $(x-1)$ divides $x^3-1$ exactly.

Flashcard 8: What is the purpose of completing the square in integration?

Answer: To transform a quadratic expression for easier integration. Creates standard forms like $(x-h)^2 + k$ for known integral formulas.

Flashcard 9: What integral technique uses $u$ substitution on quadratics?

Answer: Completing the square first. Complete the square first, then substitute $u = x + h$.

Flashcard 10: Complete the square for $x^2 + 2x + 1$. What is the result?

Answer: $(x+1)^2$. Already a perfect square, no completing needed.

Flashcard 11: Which function form is achieved by completing the square?

Answer: $(x-h)^2 + k$. This vertex form enables use of arctangent or logarithmic integrals.

Flashcard 12: Perform long division: $\frac{x^3 + 2x^2 + x + 1}{x + 1}$.

Answer: $x^2 + x + 1 + \frac{0}{x+1}$. Divide $x^3 + 2x^2 + x + 1$ by $x + 1$ step by step.

Flashcard 13: Which integral results from $\frac{1}{x^2+1}$?

Answer: $\text{arctan}(x) + C$. Standard arctangent integral form.

Flashcard 14: Perform polynomial division: $\frac{x^3 - 6x^2 + 11x - 6}{x - 2}$.

Answer: $x^2 - 4x + 3$. Synthetic or long division of cubic by linear factor.

Flashcard 15: What is the first step in integrating using long division?

Answer: Divide the numerator by the denominator. This creates polynomial and remainder terms for separate integration.

Flashcard 16: Find the integral: $\frac{1}{(x+3)^2 + 9}$.

Answer: $\frac{1}{3} \arctan \frac{x+3}{3} + C$. Use $u = \frac{x+3}{3}$ substitution with $\frac{1}{u^2+1}$ form.

Flashcard 17: When is completing the square unnecessary in integration?

Answer: When the quadratic is already a perfect square. No algebraic manipulation needed when already in standard form.

Flashcard 18: What result is obtained by integrating $\frac{1}{a^2 + x^2}$?

Answer: $\frac{1}{a} \arctan(\frac{x}{a}) + C$. Factor out $a^2$ to get $\frac{1}{a^2}\cdot\frac{1}{1+(x/a)^2}$ form.

Flashcard 19: What is the first step in integrating $\frac{1}{x^2 - 4x + 5}$?

Answer: Complete the square: $(x-2)^2 + 1$. Transform to $(x-2)^2 + 1$ for arctangent integration.

Flashcard 20: When dividing $x^3 - 1$ by $x - 1$, what do you get?

Answer: $x^2 + x + 1$. Factor theorem: $(x-1)$ divides $x^3-1$ exactly.

Flashcard 21: Perform long division: $\frac{x^3 + 2x^2 + x + 1}{x + 1}$.

Answer: $x^2 + x + 1 + \frac{0}{x+1}$. Divide $x^3 + 2x^2 + x + 1$ by $x + 1$ step by step.

Flashcard 22: Integrate $\frac{1}{x^2 - 4}$. What method simplifies it?

Answer: Partial fraction decomposition. Factor as $(x-2)(x+2)$ for partial fraction decomposition.

Flashcard 23: What technique helps simplify integration of rational functions?

Answer: Polynomial long division. Separates improper fractions into polynomial plus proper fraction.

Flashcard 24: Integrate $\frac{1}{(x-2)^2 + 4}$. What is the result?

Answer: $\frac{1}{2} \text{arctan} \frac{x-2}{2} + C$. Use $u = \frac{x-2}{2}$ substitution with $\frac{1}{u^2+1}$ form.

Flashcard 25: What is the first step in integrating $\frac{1}{x^2 - 4x + 5}$?

Answer: Complete the square: $(x-2)^2 + 1$. Transform to $(x-2)^2 + 1$ for arctangent integration.

Flashcard 26: Complete the square for $x^2 + 4x + 7$. What is the result?

Answer: $(x+2)^2 + 3$. Take half of $x$ coefficient, square it: $(\frac{4}{2})^2 = 4$, then $7-4=3$.

Flashcard 27: Perform long division: $\frac{x^4 + x^3 + x + 1}{x^2 + 1}$.

Answer: $x^2 + 1 + \frac{x}{x^2+1}$. Divide quartic by quadratic, getting quotient plus remainder fraction.

Flashcard 28: Which integral results from $\frac{1}{x^2+1}$?

Answer: $\arctan(x) + C$. Standard arctangent integral form.

Flashcard 29: What is the purpose of completing the square in integration?

Answer: To transform a quadratic expression for easier integration. Creates standard forms like $(x-h)^2 + k$ for known integral formulas.

Flashcard 30: What integral results from $\frac{1}{(x-h)^2 + k}$?

Answer: $\frac{1}{\text{sqrt}(k)} \text{arctan} \frac{x-h}{\text{sqrt}(k)} + C$. General arctangent integral formula after completing the square.

Flashcard 31: Find and correct the error: $x^2 + 6x + 8 = (x+3)^2 + 1$

Answer: Correct: $x^2 + 6x + 8 = (x+3)^2 - 1$. The constant term is $8 = 9 - 1$, not $9 + 1$.

Flashcard 32: What is the integral of $1/x$?

Answer: $\text{ln}|x| + C$. The antiderivative of the reciprocal function.

Flashcard 33: Integrate $\frac{1}{(x-2)^2 + 4}$. What is the result?

Answer: $\frac{1}{2} \text{arctan} \frac{x-2}{2} + C$. Use $u = \frac{x-2}{2}$ substitution with $\frac{1}{u^2+1}$ form.

Flashcard 34: What is the integral of $\frac{1}{x^2 - 9}$?

Answer: $\frac{1}{6} \ln\left| \frac{x-3}{x+3} \right| + C$. Partial fraction decomposition of $\frac{1}{(x-3)(x+3)}$.

Flashcard 35: What is the integral of $\frac{x}{x^2 + 4}$?

Answer: $\frac{1}{2} \text{ln}|x^2+4| + C$. Use $u$-substitution with $u = x^2 + 4$.

Flashcard 36: What technique helps simplify integration of rational functions?

Answer: Polynomial long division. Separates improper fractions into polynomial plus proper fraction.

Flashcard 37: Find and correct the error: $x^2 + 6x + 8 = (x+3)^2 + 1$

Answer: Correct: $x^2 + 6x + 8 = (x+3)^2 - 1$. The constant term is $8 = 9 - 1$, not $9 + 1$.

Flashcard 38: How can the integral $\frac{1}{x^2+6x+13}$ be simplified?

Answer: Complete the square: $(x+3)^2 + 4$. Transforms $x^2+6x+13$ into $(x+3)^2+4$ for arctangent form.

Flashcard 39: Complete the square: $4x^2 - 12x + 9$. What is the result?

Answer: $(2x-3)^2$. Recognize $4x^2 - 12x + 9$ as a perfect square trinomial.

Flashcard 40: What is the integral of $\frac{x}{x^2+1}$?

Answer: $\frac{1}{2} \text{ln}|x^2+1| + C$. $u$-substitution with $u = x^2 + 1$ gives $\frac{1}{2}\ln|u|$.

Flashcard 41: What is the benefit of transforming a quadratic in integration?

Answer: Simplifies the integral into standard forms. Standard forms match known antiderivative formulas.

Flashcard 42: How do you identify when to use long division in integration?

Answer: When the degree of the numerator is at least that of the denominator. Higher degree numerators require polynomial division before integration.

Flashcard 43: Complete the square for $x^2 + 2x + 1$. What is the result?

Answer: $(x+1)^2$. Already a perfect square, no completing needed.

Flashcard 44: Complete the square for $x^2 + 4x + 7$. What is the result?

Answer: $(x+2)^2 + 3$. Take half of $x$ coefficient, square it: $(\frac{4}{2})^2 = 4$, then $7-4=3$.

Flashcard 45: Complete the square for $x^2 - 4x + 7$. What is the result?

Answer: $(x-2)^2 + 3$. Complete the square: $(-4/2)^2 = 4$, so $7-4=3$.

Flashcard 46: Which function form is achieved by completing the square?

Answer: $(x-h)^2 + k$. This vertex form enables use of arctangent or logarithmic integrals.

Flashcard 47: Perform long division: $\frac{x^4 + x^3 + x + 1}{x^2 + 1}$.

Answer: $x^2 + 1 + \frac{x}{x^2+1}$. Divide quartic by quadratic, getting quotient plus remainder fraction.

Flashcard 48: Complete the square: $4x^2 - 12x + 9$. What is the result?

Answer: $(2x-3)^2$. Recognize $4x^2 - 12x + 9$ as a perfect square trinomial.

Flashcard 49: Perform polynomial division on $x^3 + 3x^2 + 3x + 1$ by $x + 1$.

Answer: $x^2 + 2x + 1 + \frac{0}{x+1}$. Recognize this as $(x+1)^3$ expanded, divides evenly.

Flashcard 50: How do you integrate $\frac{x^2 - 2x + 3}{x - 1}$ using long division?

Answer: Divide, then integrate the result. Long division separates into polynomial plus simple fraction terms.

Flashcard 51: Perform polynomial division on $x^3 + 3x^2 + 3x + 1$ by $x + 1$.

Answer: $x^2 + 2x + 1 + \frac{0}{x+1}$. Recognize this as $(x+1)^3$ expanded, divides evenly.

Flashcard 52: Find the integral: $\frac{1}{(x+3)^2 + 9}$.

Answer: $\frac{1}{3} \text{arctan} \frac{x+3}{3} + C$. Use $u = \frac{x+3}{3}$ substitution with $\frac{1}{u^2+1}$ form.

Flashcard 53: Why is completing the square helpful in integration?

Answer: Transforms to a form suitable for standard integrals. Creates forms matching known integral formulas like $\frac{1}{u^2+a^2}$.

Flashcard 54: How do you integrate $\frac{x^2 - 2x + 3}{x - 1}$ using long division?

Answer: Divide, then integrate the result. Long division separates into polynomial plus simple fraction terms.

Flashcard 55: What is the integral of $\frac{1}{x^2 - 9}$?

Answer: $\frac{1}{6} \text{ln}|\frac{x-3}{x+3}| + C$. Partial fraction decomposition of $\frac{1}{(x-3)(x+3)}$.

Flashcard 56: What is the integral of $\frac{x}{x^2 + 4}$?

Answer: $\frac{1}{2} \text{ln}|x^2+4| + C$. Use $u$-substitution with $u = x^2 + 4$.

Flashcard 57: What is the first step in integrating using long division?

Answer: Divide the numerator by the denominator. This creates polynomial and remainder terms for separate integration.

Flashcard 58: What integral results from $\frac{1}{(x-h)^2 + k}$?

Answer: $\frac{1}{\sqrt{k}} \arctan \frac{x-h}{\sqrt{k}} + C$. General arctangent integral formula after completing the square.

Flashcard 59: How can the integral $\frac{1}{x^2+6x+13}$ be simplified?

Answer: Complete the square: $(x+3)^2 + 4$. Transforms $x^2+6x+13$ into $(x+3)^2+4$ for arctangent form.

Flashcard 60: What result is obtained by integrating $\frac{1}{a^2 + x^2}$?

Answer: $\frac{1}{a} \text{arctan}(\frac{x}{a}) + C$. Factor out $a^2$ to get $\frac{1}{a^2}\cdot\frac{1}{1+(x/a)^2}$ form.

Flashcard 61: How do you identify when to use long division in integration?

Answer: When the degree of the numerator is at least that of the denominator. Higher degree numerators require polynomial division before integration.

Flashcard 62: Complete the square for $x^2 - 4x + 7$. What is the result?

Answer: $(x-2)^2 + 3$. Complete the square: $(-4/2)^2 = 4$, so $7-4=3$.

Flashcard 63: When is completing the square unnecessary in integration?

Answer: When the quadratic is already a perfect square. No algebraic manipulation needed when already in standard form.

Flashcard 64: What is the integral of $\frac{x}{x^2+1}$?

Answer: $\frac{1}{2} \text{ln}|x^2+1| + C$. $u$-substitution with $u = x^2 + 1$ gives $\frac{1}{2}\ln|u|$.