Integrating, Long Division, Completing the Square - AP Calculus BC
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Integrate $\frac{1}{x^2 - 4}$. What method simplifies it?
Integrate $\frac{1}{x^2 - 4}$. What method simplifies it?
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Partial fraction decomposition. Factor as $ (x-2)(x+2) $ for partial fraction decomposition.
Partial fraction decomposition. Factor as $ (x-2)(x+2) $ for partial fraction decomposition.
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What result is obtained by integrating $\frac{1}{a^2 + x^2}$?
What result is obtained by integrating $\frac{1}{a^2 + x^2}$?
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$\frac{1}{a} \arctan(\frac{x}{a}) + C$. Factor out $a^2$ to get $\frac{1}{a^2}\cdot\frac{1}{1+(x/a)^2}$ form.
$\frac{1}{a} \arctan(\frac{x}{a}) + C$. Factor out $a^2$ to get $\frac{1}{a^2}\cdot\frac{1}{1+(x/a)^2}$ form.
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What is the first step in integrating using long division?
What is the first step in integrating using long division?
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Divide the numerator by the denominator. This creates polynomial and remainder terms for separate integration.
Divide the numerator by the denominator. This creates polynomial and remainder terms for separate integration.
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Complete the square: $4x^2 - 12x + 9$. What is the result?
Complete the square: $4x^2 - 12x + 9$. What is the result?
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$(2x-3)^2$. Recognize $4x^2 - 12x + 9$ as a perfect square trinomial.
$(2x-3)^2$. Recognize $4x^2 - 12x + 9$ as a perfect square trinomial.
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Perform polynomial division: $\frac{x^3 - 6x^2 + 11x - 6}{x - 2}$.
Perform polynomial division: $\frac{x^3 - 6x^2 + 11x - 6}{x - 2}$.
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$x^2 - 4x + 3$. Synthetic or long division of cubic by linear factor.
$x^2 - 4x + 3$. Synthetic or long division of cubic by linear factor.
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What is the purpose of completing the square in integration?
What is the purpose of completing the square in integration?
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To transform a quadratic expression for easier integration. Creates standard forms like $(x-h)^2 + k$ for known integral formulas.
To transform a quadratic expression for easier integration. Creates standard forms like $(x-h)^2 + k$ for known integral formulas.
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Find and correct the error: $x^2 + 6x + 8 = (x+3)^2 + 1$
Find and correct the error: $x^2 + 6x + 8 = (x+3)^2 + 1$
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Correct: $x^2 + 6x + 8 = (x+3)^2 - 1$. The constant term is $8 = 9 - 1$, not $9 + 1$.
Correct: $x^2 + 6x + 8 = (x+3)^2 - 1$. The constant term is $8 = 9 - 1$, not $9 + 1$.
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Complete the square for $x^2 + 2x + 1$. What is the result?
Complete the square for $x^2 + 2x + 1$. What is the result?
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$(x+1)^2$. Already a perfect square, no completing needed.
$(x+1)^2$. Already a perfect square, no completing needed.
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Perform long division: $\frac{x^4 + x^3 + x + 1}{x^2 + 1}$.
Perform long division: $\frac{x^4 + x^3 + x + 1}{x^2 + 1}$.
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$x^2 + 1 + \frac{x}{x^2+1}$. Divide quartic by quadratic, getting quotient plus remainder fraction.
$x^2 + 1 + \frac{x}{x^2+1}$. Divide quartic by quadratic, getting quotient plus remainder fraction.
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How do you integrate $\frac{x^2 - 2x + 3}{x - 1}$ using long division?
How do you integrate $\frac{x^2 - 2x + 3}{x - 1}$ using long division?
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Divide, then integrate the result. Long division separates into polynomial plus simple fraction terms.
Divide, then integrate the result. Long division separates into polynomial plus simple fraction terms.
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What integral results from $\frac{1}{(x-h)^2 + k}$?
What integral results from $\frac{1}{(x-h)^2 + k}$?
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$\frac{1}{\text{sqrt}(k)} \text{arctan} \frac{x-h}{\text{sqrt}(k)} + C$. General arctangent integral formula after completing the square.
$\frac{1}{\text{sqrt}(k)} \text{arctan} \frac{x-h}{\text{sqrt}(k)} + C$. General arctangent integral formula after completing the square.
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Which integral results from $\frac{1}{x^2+1}$?
Which integral results from $\frac{1}{x^2+1}$?
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$\text{arctan}(x) + C$. Standard arctangent integral form.
$\text{arctan}(x) + C$. Standard arctangent integral form.
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Perform polynomial division on $x^3 + 3x^2 + 3x + 1$ by $x + 1$.
Perform polynomial division on $x^3 + 3x^2 + 3x + 1$ by $x + 1$.
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$x^2 + 2x + 1 + \frac{0}{x+1}$. Recognize this as $(x+1)^3$ expanded, divides evenly.
$x^2 + 2x + 1 + \frac{0}{x+1}$. Recognize this as $(x+1)^3$ expanded, divides evenly.
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Why is completing the square helpful in integration?
Why is completing the square helpful in integration?
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Transforms to a form suitable for standard integrals. Creates forms matching known integral formulas like $\frac{1}{u^2+a^2}$.
Transforms to a form suitable for standard integrals. Creates forms matching known integral formulas like $\frac{1}{u^2+a^2}$.
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What is the integral of $\frac{1}{x^2 - 9}$?
What is the integral of $\frac{1}{x^2 - 9}$?
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$\frac{1}{6} \ln\left| \frac{x-3}{x+3} \right| + C$. Partial fraction decomposition of $\frac{1}{(x-3)(x+3)}$.
$\frac{1}{6} \ln\left| \frac{x-3}{x+3} \right| + C$. Partial fraction decomposition of $\frac{1}{(x-3)(x+3)}$.
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Find the integral: $\frac{1}{(x+3)^2 + 9}$.
Find the integral: $\frac{1}{(x+3)^2 + 9}$.
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$\frac{1}{3} \arctan \frac{x+3}{3} + C$. Use $u = \frac{x+3}{3}$ substitution with $\frac{1}{u^2+1}$ form.
$\frac{1}{3} \arctan \frac{x+3}{3} + C$. Use $u = \frac{x+3}{3}$ substitution with $\frac{1}{u^2+1}$ form.
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Perform long division: $\frac{x^3 + 2x^2 + x + 1}{x + 1}$.
Perform long division: $\frac{x^3 + 2x^2 + x + 1}{x + 1}$.
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$x^2 + x + 1 + \frac{0}{x+1}$. Divide $x^3 + 2x^2 + x + 1$ by $x + 1$ step by step.
$x^2 + x + 1 + \frac{0}{x+1}$. Divide $x^3 + 2x^2 + x + 1$ by $x + 1$ step by step.
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Which function form is achieved by completing the square?
Which function form is achieved by completing the square?
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$(x-h)^2 + k$. This vertex form enables use of arctangent or logarithmic integrals.
$(x-h)^2 + k$. This vertex form enables use of arctangent or logarithmic integrals.
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How do you identify when to use long division in integration?
How do you identify when to use long division in integration?
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When the degree of the numerator is at least that of the denominator. Higher degree numerators require polynomial division before integration.
When the degree of the numerator is at least that of the denominator. Higher degree numerators require polynomial division before integration.
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What integral technique uses $u$ substitution on quadratics?
What integral technique uses $u$ substitution on quadratics?
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Completing the square first. Complete the square first, then substitute $u = x + h$.
Completing the square first. Complete the square first, then substitute $u = x + h$.
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What is the integral of $\frac{x}{x^2+1}$?
What is the integral of $\frac{x}{x^2+1}$?
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$\frac{1}{2} \text{ln}|x^2+1| + C$. $u$-substitution with $u = x^2 + 1$ gives $\frac{1}{2}\ln|u|$.
$\frac{1}{2} \text{ln}|x^2+1| + C$. $u$-substitution with $u = x^2 + 1$ gives $\frac{1}{2}\ln|u|$.
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How can the integral $\frac{1}{x^2+6x+13}$ be simplified?
How can the integral $\frac{1}{x^2+6x+13}$ be simplified?
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Complete the square: $(x+3)^2 + 4$. Transforms $x^2+6x+13$ into $(x+3)^2+4$ for arctangent form.
Complete the square: $(x+3)^2 + 4$. Transforms $x^2+6x+13$ into $(x+3)^2+4$ for arctangent form.
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What is the benefit of transforming a quadratic in integration?
What is the benefit of transforming a quadratic in integration?
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Simplifies the integral into standard forms. Standard forms match known antiderivative formulas.
Simplifies the integral into standard forms. Standard forms match known antiderivative formulas.
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What is the first step in integrating $\frac{1}{x^2 - 4x + 5}$?
What is the first step in integrating $\frac{1}{x^2 - 4x + 5}$?
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Complete the square: $(x-2)^2 + 1$. Transform to $(x-2)^2 + 1$ for arctangent integration.
Complete the square: $(x-2)^2 + 1$. Transform to $(x-2)^2 + 1$ for arctangent integration.
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Integrate $\frac{1}{(x-2)^2 + 4}$. What is the result?
Integrate $\frac{1}{(x-2)^2 + 4}$. What is the result?
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$\frac{1}{2} \text{arctan} \frac{x-2}{2} + C$. Use $u = \frac{x-2}{2}$ substitution with $\frac{1}{u^2+1}$ form.
$\frac{1}{2} \text{arctan} \frac{x-2}{2} + C$. Use $u = \frac{x-2}{2}$ substitution with $\frac{1}{u^2+1}$ form.
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What technique helps simplify integration of rational functions?
What technique helps simplify integration of rational functions?
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Polynomial long division. Separates improper fractions into polynomial plus proper fraction.
Polynomial long division. Separates improper fractions into polynomial plus proper fraction.
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When is completing the square unnecessary in integration?
When is completing the square unnecessary in integration?
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When the quadratic is already a perfect square. No algebraic manipulation needed when already in standard form.
When the quadratic is already a perfect square. No algebraic manipulation needed when already in standard form.
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Complete the square for $x^2 - 4x + 7$. What is the result?
Complete the square for $x^2 - 4x + 7$. What is the result?
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$(x-2)^2 + 3$. Complete the square: $(-4/2)^2 = 4$, so $7-4=3$.
$(x-2)^2 + 3$. Complete the square: $(-4/2)^2 = 4$, so $7-4=3$.
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When dividing $x^3 - 1$ by $x - 1$, what do you get?
When dividing $x^3 - 1$ by $x - 1$, what do you get?
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$x^2 + x + 1$. Factor theorem: $(x-1)$ divides $x^3-1$ exactly.
$x^2 + x + 1$. Factor theorem: $(x-1)$ divides $x^3-1$ exactly.
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What is the integral of $\frac{x}{x^2 + 4}$?
What is the integral of $\frac{x}{x^2 + 4}$?
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$\frac{1}{2} \text{ln}|x^2+4| + C$. Use $u$-substitution with $u = x^2 + 4$.
$\frac{1}{2} \text{ln}|x^2+4| + C$. Use $u$-substitution with $u = x^2 + 4$.
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