AP Calculus AB Quiz: Integrating Long Division Completing The Square

What this quiz covers

This quiz focuses on Integrating Long Division Completing The Square, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $\frac{x^2}{2}+4x+4\ln|x+4|+C$ (correct answer)
2. $\frac{x^2}{2}+4x+\ln|x+4|+C$
3. $\frac{x^2}{2}+4x+4\ln|x-4|+C$
4. $\frac{x^2+8x+20}{x+4}+C$
5. $\frac{x^2}{2}+4x-\frac{4}{x+4}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 2

1. $\frac{x^2}{2}+3x+11\ln|x-3|+C$ (correct answer)
2. $\frac{x^2}{2}+3x+2\ln|x-3|+C$
3. $\frac{x^2+2}{x-3}+C$
4. $\frac{x^2}{2}+3x-\frac{11}{x-3}+C$
5. $\frac{x^2}{2}+3x+11\ln|x+3|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 2 instead of 11 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 3

1. $\frac{x^2}{2}+x+\ln|x|+C$ (correct answer)
2. $\frac{x^2+x+1}{x}+C$
3. $\frac{x^2}{2}+\ln|x|+C$
4. $\frac{x^2}{2}+x+\ln|x-1|+C$
5. $\frac{x^3}{3}+\frac{x^2}{2}+x+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice C fails because it omits the linear term from the quotient, leading to an incomplete antiderivative. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 4

1. $\frac{x^2}{2}+x+10\ln|x-1|+C$ (correct answer)
2. $\frac{x^2}{2}+x+9\ln|x-1|+C$
3. $\frac{x^2+9}{x-1}+C$
4. $\frac{x^2}{2}+x-\frac{10}{x-1}+C$
5. $\frac{x^2}{2}+x+10\ln|x+1|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 9 instead of 10 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 5

1. $\frac{t^2}{2}-2t+4\ln|t-2|+C$ (correct answer)
2. $\frac{t^2}{2}-2t+\ln|t-2|+C$
3. $\frac{t^2-4t+8}{t-2}+C$
4. $\frac{t^2}{2}-2t-\frac{4}{t-2}+C$
5. $\frac{t^2}{2}-2t+4\ln|t+2|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 6

1. $\frac{x^2}{2}+5x+5\ln|x+5|+C$ (correct answer)
2. $\frac{x^2}{2}+5x+\ln|x+5|+C$
3. $\frac{x^2+10x+30}{x+5}+C$
4. $\frac{x^2}{2}+5x+5\ln|x-5|+C$
5. $\frac{x^2}{2}+5x-\frac{5}{x+5}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 5 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 7

1. $\frac{x^2}{2}+x+9\ln|x+1|+C$ (correct answer)
2. $\frac{x^2}{2}+x+10\ln|x+1|+C$
3. $\frac{x^2+2x+10}{x+1}+C$
4. $\frac{x^2}{2}+x-\frac{9}{x+1}+C$
5. $\frac{x^2}{2}+x+9\ln|x-1|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 10 instead of 9 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 8

1. $\dfrac{x^2}{2}+2x-4\ln|x-2|+C$
2. $\dfrac{x^3}{3}-4x+C$
3. $\dfrac{x^2}{2}+2x+4\ln|x-2|+C$
4. $\dfrac{x^2}{2}+2x+C$ (correct answer)
5. $\dfrac{x^2}{2}-2x+C$

Explanation: This problem demonstrates the importance of algebraic simplification before integration. Notice that $x^2-4 = (x-2)(x+2)$, so the fraction $(x^2-4)/(x-2) = (x-2)(x+2)/(x-2) = x+2$ after cancellation. This algebraic preparation transforms a seemingly complex rational function into a simple polynomial that integrates to $\frac{x^2}{2}+2x+C$. Students might mistakenly perform long division and get a logarithmic term (choices A or C), not recognizing the perfect factorization. Always check for factorization opportunities that simplify rational expressions before resorting to long division in integration problems.

Question 9

1. $\dfrac{x^2}{2}+x+\ln|x+1|+C$
2. $\dfrac{x^2}{2}+x-\ln|x+1|+C$ (correct answer)
3. $\dfrac{x^3}{3}+x^2+C$
4. $\dfrac{x^2}{2}+x-\dfrac{1}{x+1}+C$
5. $\dfrac{x^2}{2}+x+\dfrac{1}{x+1}+C$

Explanation: This problem requires polynomial long division as essential preparation. Dividing $x^2+2x$ by $x+1$ gives quotient $x+1$ with remainder $-1$, so $(x^2+2x)/(x+1) = x+1-1/(x+1)$. This algebraic step allows us to integrate: $\int(x+1)dx - \int\frac{1}{x+1}dx = \frac{x^2}{2}+x-\ln|x+1|+C$. Without division, you might incorrectly choose option A with a positive logarithm, missing how the remainder creates a subtraction. When integrating rational functions, polynomial long division separates the expression into manageable pieces that follow standard integration formulas.

Question 10

1. $\frac{x^2}{2}-x+\ln|x+1|+C$
2. $\frac{x^2}{2}-x+C$ (correct answer)
3. $\frac{x^2-1}{x+1}+C$
4. $\frac{x^2}{2}-x-\ln|x+1|+C$
5. $\frac{x^2}{2}-x+\ln|x-1|+C$

Explanation: This problem requires performing polynomial long division or simplification as preparatory algebra before integrating the rational function. The algebraic step is necessary because the rational function can be simplified by factoring and canceling common terms. To integrate, we must first simplify to a polynomial expression. This allows us to integrate the resulting polynomial directly without fractional terms. A tempting distractor like choice A fails because it includes an unnecessary logarithmic term, as if no cancellation occurred. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 11

1. $\frac{x^2}{2}+6x+4\ln|x+6|+C$ (correct answer)
2. $\frac{x^2}{2}+6x+\ln|x+6|+C$
3. $\frac{x^2}{2}+6x+4\ln|x-6|+C$
4. $\frac{x^2+12x+40}{x+6}+C$
5. $\frac{x^2}{2}+6x-\frac{4}{x+6}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 12

1. $\frac{x^2}{2}+4\ln|x|+C$ (correct answer)
2. $\frac{x^2+4}{x}+C$
3. $\frac{x^2}{2}+\ln|x|+C$
4. $\frac{x^2}{2}-4\ln|x|+C$
5. $\frac{x^3}{3}+4x+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice C fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 13

1. $\frac{x^2+1}{x-1}+C$
2. $\frac{x^2}{2}+x+\ln|x-1|+C$
3. $\frac{x^2}{2}+x+2\ln|x-1|+C$ (correct answer)
4. $\frac{x^2}{2}+x+\frac{2}{x-1}+C$
5. $\frac{x^2}{2}+x+\ln|x+1|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 2 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 14

1. $\frac{x^2}{2}+7x+11\ln|x+7|+C$ (correct answer)
2. $\frac{x^2}{2}+7x+\ln|x+7|+C$
3. $\frac{x^2+14x+60}{x+7}+C$
4. $\frac{x^2}{2}+7x+11\ln|x-7|+C$
5. $\frac{x^2}{2}+7x-\frac{11}{x+7}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 11 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 15

1. $\frac{1}{2}\arctan\!\left(\frac{x+2}{2}\right)+C$ (correct answer)
2. $\arctan\!\left(\frac{x+2}{2}\right)+C$
3. $\ln(x^2+4x+8)+C$
4. $\frac{1}{x^2+4x+8}+C$
5. $\frac{1}{2}\ln\!\left|\frac{x+2}{2}\right|+C$

Explanation: The key skill here is completing the square as preparatory algebra for integrating rational functions with quadratic denominators. For the integrand $\frac{1}{x^2 + 4x + 8}$, the quadratic does not factor over the reals since the discriminant is negative. Completing the square rewrites it as $(x + 2)^2 + 4$, which matches the form $u^2 + a^2$ with $a = 2$. This enables the use of the integral formula $\int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan\left( \frac{u}{a} \right) + C$, resulting in $\frac{1}{2} \arctan\left( \frac{x + 2}{2} \right) + C$. A tempting distractor is $\ln(x^2 + 4x + 8) + C$, but it fails because logarithmic antiderivatives apply to factorable quadratics, not this irreducible case. Always compute the discriminant of the quadratic denominator first; if negative, complete the square and apply the arctangent formula for accurate integration.

Question 16

1. $\frac{x^2}{2}-4x+9\ln|x-4|+C$ (correct answer)
2. $\frac{x^2}{2}-4x+\ln|x-4|+C$
3. $\frac{x^2}{2}-4x+9\ln|x+4|+C$
4. $\frac{x^2-8x+25}{x-4}+C$
5. $\frac{x^2}{2}-4x-\frac{9}{x-4}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 9 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 17

1. $\frac{x^2}{2}-x+4\ln|x-1|+C$ (correct answer)
2. $\frac{x^2}{2}-x+\ln|x-1|+C$
3. $\frac{x^2-2x+5}{x-1}+C$
4. $\frac{x^2}{2}-x-\frac{4}{x-1}+C$
5. $\frac{x^2}{2}-x+4\ln|x+1|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 18

1. $\frac{x^2}{2}+3x-\ln|x+4|+C$
2. $\frac{x^2}{2}+3x+3\ln|x+4|+C$ (correct answer)
3. $\frac{x^2+7x+15}{x+4}+C$
4. $\frac{x^2}{2}+3x+\ln|x+4|+C$
5. $\frac{x^2}{2}+3x+3\ln|x-4|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice A fails because it uses a negative coefficient for the logarithmic term, which does not match the positive remainder. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 19

1. $\frac{x^2}{2}+2x+20\ln|x-2|+C$ (correct answer)
2. $\frac{x^2}{2}+2x+16\ln|x-2|+C$
3. $\frac{x^2+16}{x-2}+C$
4. $\frac{x^2}{2}+2x-\frac{20}{x-2}+C$
5. $\frac{x^2}{2}+2x+20\ln|x+2|+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 16 instead of 20 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.

Question 20

1. $\frac{x^2}{2}+3x+4\ln|x+3|+C$ (correct answer)
2. $\frac{x^2}{2}+3x+\ln|x+3|+C$
3. $\frac{x^2}{2}+3x+4\ln|x-3|+C$
4. $\frac{x^2+6x+13}{x+3}+C$
5. $\frac{x^2}{2}+3x-\frac{4}{x+3}+C$

Explanation: This problem requires performing polynomial long division as preparatory algebra before integrating the rational function. The algebraic step is necessary because the degree of the numerator is equal to the degree of the denominator plus one, making the rational function improper. To integrate, we must first divide to express it as a polynomial plus a proper fraction. This allows us to integrate the polynomial terms directly and use logarithmic integration for the fractional remainder. A tempting distractor like choice B fails because it uses a coefficient of 1 instead of 4 for the logarithmic term, resulting from an incorrect remainder in the division. Always perform long division or simplify rational functions before integration to ensure accurate antiderivatives.