Integrating, Long Division, Completing the Square - AP Calculus AB

1. During a lab, the accumulated charge is modeled by $\int \frac{x^2+1}{x-1}\,dx$. Which is an antiderivative?
   Answer: $\frac{x^2}{2}+x+2\ln|x-1|+C$ 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.
2. A signal filter uses $\displaystyle \int \frac{1}{x^2-4x+8}\,dx$. What is an antiderivative?​
   Answer: $\dfrac{1}{2}\arctan\!\left(\dfrac{x-2}{2}\right)+C$ This problem requires completing the square to prepare for integration. We transform $x^2-4x+8 = (x-2)^2+4$ by completing the square: $(x-2)^2-4+8 = (x-2)^2+4$. This reveals the integral as a standard arctangent form: $\int\frac{1}{(x-2)^2+4}dx = \int\frac{1}{4[(x-2)^2/4+1]}dx = \frac{1}{2}\arctan\left(\frac{x-2}{2}\right)+C$. Students might incorrectly choose the logarithmic option (C) thinking all quadratic denominators yield logarithms, but irreducible quadratics lead to arctangent. When integrating rational functions with quadratic denominators, complete the square first to identify whether you need arctangent or logarithm formulas.
3. A rate equation uses $\int \frac{x^2-2x+5}{x-1}\,dx$. Which is an antiderivative?
   Answer: $\frac{x^2}{2}-x+4\ln|x-1|+C$ 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.
4. A fluid model uses $\int \frac{x^2-6x+20}{x-3}\,dx$. Which is an antiderivative?
   Answer: $\frac{x^2}{2}-3x+11\ln|x-3|+C$ 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.
5. A water-flow model uses $\displaystyle \int \frac{x^2+2x}{x+1}\,dx$. Find an antiderivative.​
   Answer: $\dfrac{x^2}{2}+x-\ln|x+1|+C$ 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.
6. A cost model involves $\int \frac{x^2+5x+9}{x+2}\,dx$. Which is an antiderivative?
   Answer: $\frac{x^2}{2}+3x+3\ln|x+2|+C$ 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 3 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.
7. A particle's velocity is $v(t)=\frac{t^2-4t+8}{t-2}$. What is an antiderivative of $v(t)$?
   Answer: $\frac{t^2}{2}-2t+4\ln|t-2|+C$ 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.
8. A controller uses $\int \frac{x^2+6x+13}{x+3}\,dx$. Which is an antiderivative?
   Answer: $\frac{x^2}{2}+3x+4\ln|x+3|+C$ 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.
9. In a modeling problem, determine an antiderivative of $\int \frac{x^2+9}{x-1}\,dx$.
   Answer: $\frac{x^2}{2}+x+10\ln|x-1|+C$ 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.
10. A control system uses $\int \frac{x^2+12x+40}{x+6}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}+6x+4\ln|x+6|+C$ 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.
11. A sensor model requires $\int \frac{x^2-8x+25}{x-4}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}-4x+9\ln|x-4|+C$ 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.
12. A data-fitting step requires $\int \frac{x^2+10x+30}{x+5}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}+5x+5\ln|x+5|+C$ 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.
13. A dynamics integral is $\int \frac{x^2+16}{x-2}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}+2x+20\ln|x-2|+C$ 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.
14. In a heat-transfer model, find $\displaystyle \int \frac{1}{x^2+6x+13}\,dx$.​
    Answer: $\dfrac{1}{2}\arctan\!\left(\dfrac{x+3}{2}\right)+C$ This integral requires completing the square as preparatory algebra for integration. We rewrite $x^2+6x+13 = (x+3)^2+4$ by completing the square: add and subtract $(6/2)^2 = 9$ to get $(x+3)^2+4$. This transforms the integral into the standard arctangent form: $\int\frac{1}{(x+3)^2+4}dx = \int\frac{1}{4[(x+3)^2/4+1]}dx = \frac{1}{2}\arctan\left(\frac{x+3}{2}\right)+C$. Without completing the square, you might incorrectly attempt a logarithmic antiderivative (choice C) since the denominator is quadratic. For integrals with irreducible quadratic denominators, always complete the square to reveal the arctangent structure.
15. For a cost model, evaluate an antiderivative of $\displaystyle \int \frac{x^2-4}{x-2}\,dx$.​
    Answer: $\dfrac{x^2}{2}+2x+C$ 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.
16. A diffusion model uses $\int \frac{1}{x^2+10x+29}\,dx$. Which is an antiderivative?
    Answer: $\frac{1}{2}\arctan\!\left(\frac{x+5}{2}\right)+C$ The key skill here is completing the square as preparatory algebra for integrating rational functions with quadratic denominators. For the integrand 1/(x² + 10x + 29), the quadratic does not factor over the reals since the discriminant is negative. Completing the square rewrites it as (x + 5)² + 4, which matches the form u² + a² with a = 2. This enables the use of the integral formula ∫ du/(u² + a²) = (1/a) arctan(u/a) + C, resulting in (1/2) arctan((x + 5)/2) + C. A tempting distractor is ln(x² + 10x + 29) + 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.
17. In an optimization routine, find an antiderivative of $\int \frac{x^2+3x+10}{x+2}\,dx$.
    Answer: $\frac{x^2}{2}+x+8\ln|x+2|+C$ 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 8 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.
18. An engineering integral is $\int \frac{x^2+1}{x+1}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}-x+2\ln|x+1|+C$ 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.
19. A physics integral is $\int \frac{x^2+8x+20}{x+4}\,dx$. Which is an antiderivative?
    Answer: $\frac{x^2}{2}+4x+4\ln|x+4|+C$ 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.
20. For a growth model, find an antiderivative of $\displaystyle \int \frac{x^2+4x+5}{x+2}\,dx$.​
    Answer: $\dfrac{x^2}{2}+2x+\ln|x+2|+C$ This integration requires polynomial long division as preparatory algebra. Dividing $x^2+4x+5$ by $x+2$ yields quotient $x+2$ with remainder $1$, so $(x^2+4x+5)/(x+2) = x+2+1/(x+2)$. This separation enables term-by-term integration: $\int(x+2)dx + \int\frac{1}{x+2}dx = \frac{x^2}{2}+2x+\ln|x+2|+C$. Without long division, you might incorrectly choose option B with a negative logarithm, misunderstanding how the remainder affects the sign. Always perform long division when the numerator's degree equals or exceeds the denominator's degree to properly separate polynomial and logarithmic parts.

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