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

1. Integrate $\frac{1}{x^2 - 4}$. What method simplifies it?

   Partial fraction decomposition. Factor as $ (x-2)(x+2) $ for partial fraction decomposition.

2. What is the integral of $1/x$?

   $\text{ln}|x| + C$. The antiderivative of the reciprocal function.

3. Perform polynomial division: $\frac{x^3 - 6x^2 + 11x - 6}{x - 2}$.

   $x^2 - 4x + 3$. Synthetic or long division of cubic by linear factor.

4. What integral technique uses $u$ substitution on quadratics?

   Completing the square first. Complete the square first, then substitute $u = x + h$.

5. What is the benefit of transforming a quadratic in integration?

   Simplifies the integral into standard forms. Standard forms match known antiderivative formulas.

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

   To transform a quadratic expression for easier integration. Creates standard forms like $(x-h)^2 + k$ for known integral formulas.

7. Complete the square for $x^2 + 2x + 1$. What is the result?

   $(x+1)^2$. Already a perfect square, no completing needed.

8. Which function form is achieved by completing the square?

   $(x-h)^2 + k$. This vertex form enables use of arctangent or logarithmic integrals.

9. Perform long division: $\frac{x^3 + 2x^2 + x + 1}{x + 1}$.

   $x^2 + x + 1 + \frac{0}{x+1}$. Divide $x^3 + 2x^2 + x + 1$ by $x + 1$ step by step.

10. Which integral results from $\frac{1}{x^2+1}$?

    $\text{arctan}(x) + C$. Standard arctangent integral form.

11. What is the first step in integrating using long division?

    Divide the numerator by the denominator. This creates polynomial and remainder terms for separate integration.

12. Find the integral: $\frac{1}{(x+3)^2 + 9}$.

    $\frac{1}{3} \arctan \frac{x+3}{3} + C$. Use $u = \frac{x+3}{3}$ substitution with $\frac{1}{u^2+1}$ form.

13. When is completing the square unnecessary in integration?

    When the quadratic is already a perfect square. No algebraic manipulation needed when already in standard form.

14. What result is obtained by integrating $\frac{1}{a^2 + x^2}$?

    $\frac{1}{a} \arctan(\frac{x}{a}) + C$. Factor out $a^2$ to get $\frac{1}{a^2}\cdot\frac{1}{1+(x/a)^2}$ form.

15. When dividing $x^3 - 1$ by $x - 1$, what do you get?

    $x^2 + x + 1$. Factor theorem: $(x-1)$ divides $x^3-1$ exactly.

16. What is the first step in integrating $\frac{1}{x^2 - 4x + 5}$?

    Complete the square: $(x-2)^2 + 1$. Transform to $(x-2)^2 + 1$ for arctangent integration.

17. Perform long division: $\frac{x^4 + x^3 + x + 1}{x^2 + 1}$.

    $x^2 + 1 + \frac{x}{x^2+1}$. Divide quartic by quadratic, getting quotient plus remainder fraction.

18. What integral results from $\frac{1}{(x-h)^2 + k}$?

    $\frac{1}{\text{sqrt}(k)} \text{arctan} \frac{x-h}{\text{sqrt}(k)} + C$. General arctangent integral formula after completing the square.

19. Find and correct the error: $x^2 + 6x + 8 = (x+3)^2 + 1$

    Correct: $x^2 + 6x + 8 = (x+3)^2 - 1$. The constant term is $8 = 9 - 1$, not $9 + 1$.

20. Integrate $\frac{1}{(x-2)^2 + 4}$. What is the result?

    $\frac{1}{2} \text{arctan} \frac{x-2}{2} + C$. Use $u = \frac{x-2}{2}$ substitution with $\frac{1}{u^2+1}$ form.

21. What is the integral of $\frac{1}{x^2 - 9}$?

    $\frac{1}{6} \ln\left| \frac{x-3}{x+3} \right| + C$. Partial fraction decomposition of $\frac{1}{(x-3)(x+3)}$.

22. What is the integral of $\frac{x}{x^2 + 4}$?

    $\frac{1}{2} \text{ln}|x^2+4| + C$. Use $u$-substitution with $u = x^2 + 4$.

23. What technique helps simplify integration of rational functions?

    Polynomial long division. Separates improper fractions into polynomial plus proper fraction.

24. How can the integral $\frac{1}{x^2+6x+13}$ be simplified?

    Complete the square: $(x+3)^2 + 4$. Transforms $x^2+6x+13$ into $(x+3)^2+4$ for arctangent form.

25. Complete the square: $4x^2 - 12x + 9$. What is the result?

    $(2x-3)^2$. Recognize $4x^2 - 12x + 9$ as a perfect square trinomial.

26. How do you identify when to use long division in integration?

    When the degree of the numerator is at least that of the denominator. Higher degree numerators require polynomial division before integration.

27. Complete the square for $x^2 + 4x + 7$. What is the result?

    $(x+2)^2 + 3$. Take half of $x$ coefficient, square it: $(\frac{4}{2})^2 = 4$, then $7-4=3$.

28. Complete the square for $x^2 - 4x + 7$. What is the result?

    $(x-2)^2 + 3$. Complete the square: $(-4/2)^2 = 4$, so $7-4=3$.

29. Perform polynomial division on $x^3 + 3x^2 + 3x + 1$ by $x + 1$.

    $x^2 + 2x + 1 + \frac{0}{x+1}$. Recognize this as $(x+1)^3$ expanded, divides evenly.

30. Why is completing the square helpful in integration?

    Transforms to a form suitable for standard integrals. Creates forms matching known integral formulas like $\frac{1}{u^2+a^2}$.

31. How do you integrate $\frac{x^2 - 2x + 3}{x - 1}$ using long division?

    Divide, then integrate the result. Long division separates into polynomial plus simple fraction terms.

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

    $\frac{1}{2} \text{ln}|x^2+1| + C$. $u$-substitution with $u = x^2 + 1$ gives $\frac{1}{2}\ln|u|$.

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