Selecting Techniques for Antidifferentiation - AP Calculus AB

1. What is the antiderivative of $\text{cos}^2(x)$?

   $\frac{x}{2} + \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{cos}^2(x) = \frac{1 + \text{cos}(2x)}{2}$.

2. Choose the technique for $\text{ln}(a^x)$.

   Substitution. Simplify $\text{ln}(a^x) = x\text{ln}(a)$ before integrating.

3. What is the antiderivative of $\cot(x)$?

   $\ln |\sin(x)| + C$. Rewrite $\cot(x) = \frac{\cos(x)}{\sin(x)}$ and use substitution.

4. What is the antiderivative of $\text{cot}(x)$?

   $\text{ln}|\text{sin}(x)| + C$. Rewrite $\text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)}$ and use substitution.

5. Find the antiderivative of $xe^{x}$.

   Integration by parts. Product of polynomial and exponential requires integration by parts.

6. What is the antiderivative of $\frac{1}{\sqrt{1-x^2}}$?

   $\arcsin(x) + C$. Standard inverse sine antiderivative formula.

7. What is the antiderivative of $\frac{1}{x}$?

   $\text{ln}|x| + C$. The derivative of $\text{ln}|x|$ is $\frac{1}{x}$.

8. Find the antiderivative of $\text{sec}^2(x)$.

   $\text{tan}(x) + C$. The derivative of $\text{tan}(x)$ is $\text{sec}^2(x)$.

9. Select the technique for $e^{x^2}$.

   No elementary antiderivative. Cannot be expressed using elementary functions.

10. State the antiderivative of $\text{cos}(x)$.

    $\text{sin}(x) + C$. The derivative of $\text{sin}(x)$ is $\text{cos}(x)$.

11. What is the antiderivative of $\frac{1}{\text{sqrt}(1-x^2)}$?

    $\text{arcsin}(x) + C$. Standard inverse sine antiderivative formula.

12. What is the antiderivative of $a^x$?

    $\frac{a^x}{\text{ln}(a)} + C$. General exponential function antiderivative formula with base $a$.

13. What is the antiderivative of $e^x$?

    $e^x + C$. The derivative of $e^x$ is itself, so the antiderivative reverses this.

14. What is the antiderivative of $\text{sec}(x)\text{tan}(x)$?

    $\text{sec}(x) + C$. The derivative of $\text{sec}(x)$ is $\text{sec}(x)\text{tan}(x)$.

15. Which technique is suitable for $\frac{x}{x^2 + 1}$?

    Substitution. Use $u = x^2 + 1$ to simplify the integrand.

16. Identify the technique for $\text{arccos}(x)$.

    Integration by parts. Inverse trig functions require integration by parts technique.

17. What is the antiderivative of $\text{csc}^2(x)$?

    $-\text{cot}(x) + C$. The derivative of $-\text{cot}(x)$ is $\text{csc}^2(x)$.

18. What is the antiderivative of $\text{sin}^2(x)$?

    $\frac{x}{2} - \frac{1}{4}\text{sin}(2x) + C$. Use double angle identity: $\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}$.

19. What substitution is used for $\text{sqrt}(x^2 + a^2)$?

    Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to handle the square root expression.

20. What is the antiderivative of $x^{n}$?

    $\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1 and divides.

21. What is the antiderivative of $\text{csc}(x)\text{cot}(x)$?

    $-\text{csc}(x) + C$. The derivative of $-\text{csc}(x)$ is $\text{csc}(x)\text{cot}(x)$.

22. What is the antiderivative of $\text{arcsin}(x)$?

    $x \text{arcsin}(x) + \text{sqrt}(1 - x^2) + C$. Standard result from integration by parts formula.

23. What is the antiderivative of $\sec(x)\tan(x)$?

    $\sec(x) + C$. The derivative of $\sec(x)$ is $\sec(x)\tan(x)$.

24. Choose the integration technique for $\text{arctan}(x)$.

    Integration by parts. Inverse trig functions require integration by parts technique.

25. What is the antiderivative of $\text{tan}(x)$?

    $-\text{ln}|\text{cos}(x)| + C$. Rewrite $\text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)}$ and use substitution.

26. Identify the technique for $\frac{1}{\text{sqrt}(x^2 + a^2)}$?

    Trigonometric substitution. Use $x = a\text{tan}(\theta)$ to simplify the square root.

27. Identify the technique for $e^{3x}\text{sin}(x)$.

    Integration by parts. Product of exponential and sine requires integration by parts.

28. What is the antiderivative of $\frac{1}{x^2 + a^2}$?

    $\frac{1}{a}\arctan(\frac{x}{a}) + C$. Standard arctangent formula with scaling factor $\frac{1}{a}$.

29. State the antiderivative of $\text{sin}(x)$.

    $-\text{cos}(x) + C$. The derivative of $-\text{cos}(x)$ is $\text{sin}(x)$.

30. Identify the integration technique for $x\text{sin}(x)$.

    Integration by parts. Product of $x$ and trig function requires integration by parts.

31. Which technique is appropriate for $e^{2x}$?

    Substitution. Use $u = 2x$ substitution to handle the coefficient.

32. Identify the technique for $x^2 \text{ln}(x)$.

    Integration by parts. Polynomial times logarithm requires integration by parts.

33. Which substitution is used for $\frac{1}{\text{sqrt}(a^2 - x^2)}$?

    Trigonometric substitution. Use $x = a\text{sin}(\theta)$ to eliminate the square root.

34. Identify the integration technique for $\text{ln}(x)$.

    Integration by parts. Use $u = \text{ln}(x)$ and $dv = dx$ for integration by parts.

35. What is the antiderivative of $\frac{1}{x^2}$?

    $-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.

36. Which method is suitable for $x\text{cos}(x)$?

    Integration by parts. Product of polynomial and cosine requires integration by parts.

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