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

1. What is the completed square form of $x^2 + 2x + 3$?

   $(x+1)^2 + 2$. Complete the square: $(x+1)^2 + 2$ where $2 = 3 - 1$.

2. Express $x^2 + 4x + 13$ in completed square form.

   $(x+2)^2 + 9$. Complete the square by adding and subtracting $(\frac{4}{2})^2 = 4$.

3. State the integral of $\frac{1}{x^2 + 2x + 3}$ after completing the square.

   Use $\frac{1}{(x+1)^2 + 2}$ form. After completing the square: $x^2 + 2x + 3 = (x+1)^2 + 2$.

4. Express $x^2 + 14x + 49$ in completed square form.

   $(x+7)^2$. This is a perfect square trinomial: $(x+7)^2$.

5. What is the integral of $\frac{1}{(x+3)^2 + 1}$?

   $\tan^{-1}(x+3) + C$. Standard arctangent integral form.

6. Convert $x^2 + 12x + 36$ into a completed square.

   $(x+6)^2$. Perfect square trinomial with $a = 6$.

7. What is the integral of $\frac{1}{(x+1)^2 + 4}$?

   $\frac{1}{2} \tan^{-1}\frac{x+1}{2} + C$. Arctangent formula with $a = 2$.

8. What is the integral of $\frac{1}{(x+6)^2}$?

   $-\frac{1}{x+6} + C$. Power rule: $\int (x+6)^{-2} dx = -(x+6)^{-1} + C$.

9. What is the integral of $\frac{1}{(x+2)^2+1}$?

   $\tan^{-1}(x+2) + C$. Standard arctangent integral with $a = 1$.

10. What is the antiderivative of $\frac{1}{(x+1)^2}$?

    $-\frac{1}{x+1} + C$. Standard integral of $u^{-2}$ form.

11. Convert $x^2 + 4x + 6$ into completed square form.

    $(x+2)^2 + 2$. Complete the square: $(x+2)^2 + 2$ where $2 = 6 - 4$.

12. Convert $x^2 + 4x + 5$ into completed square form.

    $(x+2)^2 + 1$. Complete the square: $x^2 + 4x + 4 + 1 = (x+2)^2 + 1$.

13. What is the integral of $\frac{1}{(x+1)^2 + 2}$?

    $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+1}{\sqrt{2}} + C$. Use arctangent formula with $a = \sqrt{2}$.

14. Find the completed square form of $x^2 + 10x + 25$.

    $(x+5)^2$. This is already a perfect square trinomial.

15. What is the integral of $\frac{1}{(x+7)^2}$?

    $-\frac{1}{x+7} + C$. Standard power rule for $u^{-2}$

16. Express $x^2 + 2x + 2$ in completed square form.

    $(x+1)^2 + 1$. Add and subtract $(\frac{2}{2})^2 = 1$ to complete the square.

17. What is the integral of $\frac{1}{(x+2)^2 + 2}$?

    $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+2}{\sqrt{2}} + C$. Arctangent formula with $a = \sqrt{2}$.

18. What is the integral of $\frac{1}{(x+3)^2 + 2}$?

    $\frac{1}{\sqrt{2}} \tan^{-1}\frac{x+3}{\sqrt{2}} + C$. Use the formula $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.

19. What is the integral of $\frac{1}{(x+2)^2 + 4}$?

    $\frac{1}{2} \tan^{-1}\frac{x+2}{2} + C$. Use $\int \frac{1}{x^2 + a^2} dx$ formula with $a = 2$.

20. Use long division for $\frac{x^2 + 3x + 2}{x + 1}$. What is the quotient?

    $x + 2$. Long division gives quotient and remainder.

21. Perform long division on $\frac{x^2 + 5x + 6}{x + 1}$. What is the quotient?

    $x + 4$. Polynomial long division of quadratic by linear.

22. How do you handle $\frac{x^2 + 3x + 5}{x + 2}$ using long division?

    Divide $x^2 + 3x + 5$ by $x + 2$. Degree of numerator > degree of denominator requires polynomial division first.

23. Use long division for $\frac{x^3 + 6x^2 + 11x + 6}{x + 3}$. What is the quotient?

    $x^2 + 3x + 2$. Divide cubic by linear using polynomial long division.

24. What is the integral of $\frac{1}{(x+5)^2}$?

    $-\frac{1}{x+5} + C$. Integral of $u^{-2}$ is $-u^{-1} + C$.

25. What is the first step in completing the square for $x^2 + 10x + 29$?

    Write as $(x+5)^2 + 4$. Take half the coefficient of $x$: $(\frac{10}{2})^2 = 25$, then adjust.

26. What is the first step when integrating $\frac{x^3 + 2x^2 + x + 1}{x^2 + 1}$ using long division?

    Divide $x^3 + 2x^2 + x + 1$ by $x^2 + 1$. Use polynomial long division when the degree of numerator ≥ degree of denominator.

27. What is the purpose of completing the square in integration?

    To simplify the integrand into a recognizable form. Transforms expressions into standard forms that match integration formulas.

28. Find the completed square form of $x^2 + 2x + 1$.

    $(x+1)^2$. Perfect square trinomial: $(x+1)^2$.

29. Convert $x^2 + 8x + 16$ into a completed square.

    $(x+4)^2$. This is already a perfect square trinomial.

30. Convert $x^2 + 2x + 5$ into completed square form.

    $(x+1)^2 + 4$. Complete the square: add/subtract $(\frac{2}{2})^2 = 1$.

31. Identify the integral of $\frac{1}{x^2 + 4x + 13}$ after completing the square.

    Use $\frac{1}{(x+2)^2 + 9}$ form. Complete the square: $x^2 + 4x + 13 = (x+2)^2 + 9$.

32. What is the antiderivative of $\frac{1}{(x+2)^2 + 9}$?

    $\frac{1}{3} \tan^{-1} \left( \frac{x+2}{3} \right) + C$. Use arctangent formula with $a = 3$: $\frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.

33. What is the integral of $\frac{1}{(x+1)^2 + 1}$?

    $\tan^{-1}(x+1) + C$. Arctangent integral with $a = 1$.

34. Identify the completed square form of $x^2 + 6x + 11$.

    $(x + 3)^2 + 2$. Take half the coefficient of $x$, square it: $(\frac{6}{2})^2 = 9$, so add/subtract 9.

35. Express $x^2 + 6x + 10$ in completed square form.

    $ $(x+3)^2$ + 1 $. Complete the square: $ $(x+3)^2$ + 1 $ where $1 = 10 - 9$.

36. What is the completed square form of $x^2 + 4x + 8$?

    $(x+2)^2 + 4$. Complete the square: $(x+2)^2 + 4$ where $4 = 8 - 4$.

37. Perform long division on $\frac{x^2 + x + 1}{x + 1}$. What is the quotient?

    $x$. Polynomial division of $x^2 + x + 1$ by $x + 1$.

38. What is the integral of $\frac{1}{(x+4)^2}$?

    $-\frac{1}{x+4} + C$. Use the power rule: $\int u^{-2} du = -u^{-1} + C$.

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