Integrating, Long Division, Completing the Square - AP Calculus AB
Card 1 of 30
Find the completed square form of $x^2 + 2x + 1$.
Find the completed square form of $x^2 + 2x + 1$.
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$(x+1)^2$. Perfect square trinomial: $(x+1)^2$.
$(x+1)^2$. Perfect square trinomial: $(x+1)^2$.
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Convert $x^2 + 4x + 6$ into completed square form.
Convert $x^2 + 4x + 6$ into completed square form.
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$(x+2)^2 + 2$. Complete the square: $(x+2)^2 + 2$ where $2 = 6 - 4$.
$(x+2)^2 + 2$. Complete the square: $(x+2)^2 + 2$ where $2 = 6 - 4$.
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Express $x^2 + 4x + 13$ in completed square form.
Express $x^2 + 4x + 13$ in completed square form.
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$(x+2)^2 + 9$. Complete the square by adding and subtracting $(\frac{4}{2})^2 = 4$.
$(x+2)^2 + 9$. Complete the square by adding and subtracting $(\frac{4}{2})^2 = 4$.
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What is the integral of $\frac{1}{(x+2)^2 + 4}$?
What is the integral of $\frac{1}{(x+2)^2 + 4}$?
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$\frac{1}{2} \tan^{-1}\frac{x+2}{2} + C$. Use $\int \frac{1}{x^2 + a^2} dx$ formula with $a = 2$.
$\frac{1}{2} \tan^{-1}\frac{x+2}{2} + C$. Use $\int \frac{1}{x^2 + a^2} dx$ formula with $a = 2$.
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How do you handle $\frac{x^2 + 3x + 5}{x + 2}$ using long division?
How do you handle $\frac{x^2 + 3x + 5}{x + 2}$ using long division?
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Divide $x^2 + 3x + 5$ by $x + 2$. Degree of numerator > degree of denominator requires polynomial division first.
Divide $x^2 + 3x + 5$ by $x + 2$. Degree of numerator > degree of denominator requires polynomial division first.
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Perform long division on $\frac{x^2 + 5x + 6}{x + 1}$. What is the quotient?
Perform long division on $\frac{x^2 + 5x + 6}{x + 1}$. What is the quotient?
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$x + 4$. Polynomial long division of quadratic by linear.
$x + 4$. Polynomial long division of quadratic by linear.
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Identify the completed square form of $x^2 + 6x + 11$.
Identify the completed square form of $x^2 + 6x + 11$.
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$(x + 3)^2 + 2$. Take half the coefficient of $x$, square it: $(\frac{6}{2})^2 = 9$, so add/subtract 9.
$(x + 3)^2 + 2$. Take half the coefficient of $x$, square it: $(\frac{6}{2})^2 = 9$, so add/subtract 9.
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What is the completed square form of $x^2 + 2x + 3$?
What is the completed square form of $x^2 + 2x + 3$?
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$(x+1)^2 + 2$. Complete the square: $(x+1)^2 + 2$ where $2 = 3 - 1$.
$(x+1)^2 + 2$. Complete the square: $(x+1)^2 + 2$ where $2 = 3 - 1$.
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Use long division for $\frac{x^2 + 3x + 2}{x + 1}$. What is the quotient?
Use long division for $\frac{x^2 + 3x + 2}{x + 1}$. What is the quotient?
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$x + 2$. Long division gives quotient and remainder.
$x + 2$. Long division gives quotient and remainder.
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What is the antiderivative of $\frac{1}{(x+2)^2 + 9}$?
What is the antiderivative of $\frac{1}{(x+2)^2 + 9}$?
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$\frac{1}{3} \tan^{-1} \left( \frac{x+2}{3} \right) + C$. Use arctangent formula with $a = 3$: $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.
$\frac{1}{3} \tan^{-1} \left( \frac{x+2}{3} \right) + C$. Use arctangent formula with $a = 3$: $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.
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What is the integral of $\frac{1}{(x+3)^2 + 1}$?
What is the integral of $\frac{1}{(x+3)^2 + 1}$?
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$\tan^{-1}(x+3) + C$. Standard arctangent integral form.
$\tan^{-1}(x+3) + C$. Standard arctangent integral form.
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What is the completed square form of $x^2 + 4x + 8$?
What is the completed square form of $x^2 + 4x + 8$?
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$(x+2)^2 + 4$. Complete the square: $(x+2)^2 + 4$ where $4 = 8 - 4$.
$(x+2)^2 + 4$. Complete the square: $(x+2)^2 + 4$ where $4 = 8 - 4$.
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What is the integral of $\frac{1}{(x+1)^2 + 4}$?
What is the integral of $\frac{1}{(x+1)^2 + 4}$?
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$\frac{1}{2} \tan^{-1}\frac{x+1}{2} + C$. Arctangent formula with $a = 2$.
$\frac{1}{2} \tan^{-1}\frac{x+1}{2} + C$. Arctangent formula with $a = 2$.
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Express $x^2 + 2x + 2$ in completed square form.
Express $x^2 + 2x + 2$ in completed square form.
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$(x+1)^2 + 1$. Add and subtract $(\frac{2}{2})^2 = 1$ to complete the square.
$(x+1)^2 + 1$. Add and subtract $(\frac{2}{2})^2 = 1$ to complete the square.
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What is the integral of $\frac{1}{(x+5)^2}$?
What is the integral of $\frac{1}{(x+5)^2}$?
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$-\frac{1}{x+5} + C$. Integral of $u^{-2}$ is $-u^{-1} + C$.
$-\frac{1}{x+5} + C$. Integral of $u^{-2}$ is $-u^{-1} + C$.
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What is the first step in completing the square for $x^2 + 10x + 29$?
What is the first step in completing the square for $x^2 + 10x + 29$?
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Write as $(x+5)^2 + 4$. Take half the coefficient of $x$: $(\frac{10}{2})^2 = 25$, then adjust.
Write as $(x+5)^2 + 4$. Take half the coefficient of $x$: $(\frac{10}{2})^2 = 25$, then adjust.
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Use long division for $\frac{x^3 + 6x^2 + 11x + 6}{x + 3}$. What is the quotient?
Use long division for $\frac{x^3 + 6x^2 + 11x + 6}{x + 3}$. What is the quotient?
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$x^2 + 3x + 2$. Divide cubic by linear using polynomial long division.
$x^2 + 3x + 2$. Divide cubic by linear using polynomial long division.
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Convert $x^2 + 2x + 5$ into completed square form.
Convert $x^2 + 2x + 5$ into completed square form.
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$(x+1)^2 + 4$. Complete the square: add/subtract $(\frac{2}{2})^2 = 1$.
$(x+1)^2 + 4$. Complete the square: add/subtract $(\frac{2}{2})^2 = 1$.
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What is the integral of $\frac{1}{(x+6)^2}$?
What is the integral of $\frac{1}{(x+6)^2}$?
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$-\frac{1}{x+6} + C$. Power rule: $\int (x+6)^{-2} dx = -(x+6)^{-1} + C$.
$-\frac{1}{x+6} + C$. Power rule: $\int (x+6)^{-2} dx = -(x+6)^{-1} + C$.
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Convert $x^2 + 4x + 5$ into completed square form.
Convert $x^2 + 4x + 5$ into completed square form.
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$(x+2)^2 + 1$. Complete the square: $x^2 + 4x + 4 + 1 = (x+2)^2 + 1$.
$(x+2)^2 + 1$. Complete the square: $x^2 + 4x + 4 + 1 = (x+2)^2 + 1$.
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Convert $x^2 + 8x + 16$ into a completed square.
Convert $x^2 + 8x + 16$ into a completed square.
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$(x+4)^2$. This is already a perfect square trinomial.
$(x+4)^2$. This is already a perfect square trinomial.
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What is the integral of $\frac{1}{(x+3)^2 + 2}$?
What is the integral of $\frac{1}{(x+3)^2 + 2}$?
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$\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+3}{\sqrt{2}} + C$. Use the formula $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.
$\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+3}{\sqrt{2}} + C$. Use the formula $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.
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What is the integral of $\frac{1}{(x+4)^2}$?
What is the integral of $\frac{1}{(x+4)^2}$?
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$-\frac{1}{x+4} + C$. Use the power rule: $\int u^{-2} du = -u^{-1} + C$.
$-\frac{1}{x+4} + C$. Use the power rule: $\int u^{-2} du = -u^{-1} + C$.
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Perform long division on $\frac{x^2 + x + 1}{x + 1}$. What is the quotient?
Perform long division on $\frac{x^2 + x + 1}{x + 1}$. What is the quotient?
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$x$. Polynomial division of $x^2 + x + 1$ by $x + 1$.
$x$. Polynomial division of $x^2 + x + 1$ by $x + 1$.
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What is the integral of $\frac{1}{(x+1)^2 + 1}$?
What is the integral of $\frac{1}{(x+1)^2 + 1}$?
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$\tan^{-1}(x+1) + C$. Arctangent integral with $a = 1$.
$\tan^{-1}(x+1) + C$. Arctangent integral with $a = 1$.
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Find the completed square form of $x^2 + 10x + 25$.
Find the completed square form of $x^2 + 10x + 25$.
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$(x+5)^2$. This is already a perfect square trinomial.
$(x+5)^2$. This is already a perfect square trinomial.
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What is the antiderivative of $\frac{1}{(x+1)^2}$?
What is the antiderivative of $\frac{1}{(x+1)^2}$?
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$-\frac{1}{x+1} + C$. Standard integral of $u^{-2}$ form.
$-\frac{1}{x+1} + C$. Standard integral of $u^{-2}$ form.
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What is the integral of $\frac{1}{(x+2)^2+1}$?
What is the integral of $\frac{1}{(x+2)^2+1}$?
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$\tan^{-1}(x+2) + C$. Standard arctangent integral with $a = 1$.
$\tan^{-1}(x+2) + C$. Standard arctangent integral with $a = 1$.
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Convert $x^2 + 12x + 36$ into a completed square.
Convert $x^2 + 12x + 36$ into a completed square.
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$(x+6)^2$. Perfect square trinomial with $a = 6$.
$(x+6)^2$. Perfect square trinomial with $a = 6$.
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Identify the integral of $\frac{1}{x^2 + 4x + 13}$ after completing the square.
Identify the integral of $\frac{1}{x^2 + 4x + 13}$ after completing the square.
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Use $\frac{1}{(x+2)^2 + 9}$ form. Complete the square: $x^2 + 4x + 13 = (x+2)^2 + 9$.
Use $\frac{1}{(x+2)^2 + 9}$ form. Complete the square: $x^2 + 4x + 13 = (x+2)^2 + 9$.
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