Finding Antiderivatives and Indefinite Integrals - AP Calculus AB

1. What is the antiderivative of $f(x) = \sec^2(x)$?

   $F(x) = \tan(x) + C$. Derivative of $\tan(x)$ is $\sec^2(x)$, so reverse gives this.

2. Identify the rule used to find the antiderivative of $cf(x)$.

   Constant multiple rule: $F(x) = cF(x) + C$. Constants can be factored out of integrals.

3. Which rule is applied for $\frac{d}{dx} \text{F}(x) = f(x)$?

   The Fundamental Theorem of Calculus, Part 1. States that antiderivative and derivative are inverse operations.

4. What is the antiderivative of $f(x) = e^x$?

   $F(x) = e^x + C$. Exponential function $e^x$ is its own antiderivative.

5. What is the antiderivative of $f(x) = x^n$ where $n \neq -1$?

   $F(x) = \frac{x^{n+1}}{n+1} + C$. Power rule: increase exponent by 1, divide by new exponent.

6. What is the antiderivative of $f(x) = \text{cos}(x)$?

   $F(x) = \text{sin}(x) + C$. Derivative of $\sin(x)$ is $\cos(x)$, so reverse gives this.

7. Find the indefinite integral of $f(x) = 7$.

   $F(x) = 7x + C$. Antiderivative of constant $7$ is $7x$.

8. What is the antiderivative of $f(x) = \text{sec}^2(x)$?

   $F(x) = \text{tan}(x) + C$. Derivative of $\tan(x)$ is $\sec^2(x)$, so reverse gives this.

9. What is the indefinite integral of $f(x) = -9x^5$?

   $F(x) = -\frac{3}{2}x^6 + C$. Apply power rule: $\int -9x^5 dx = \frac{-9x^6}{6}$.

10. What is the indefinite integral of $f(x) = \frac{3}{x}$?

    $F(x) = 3 \ln |x| + C$. Constant multiple of $\frac{1}{x}$ antiderivative.

11. Find the indefinite integral of $f(x) = x^{-1/2}$.

    $F(x) = 2x^{1/2} + C$. Apply power rule: $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} = 2x^{1/2}$.

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

    $F(x) = -\text{cot}(x) + C$. Derivative of $-\cot(x)$ is $\csc^2(x)$, so reverse gives this.

13. Find the indefinite integral of $f(x) = 3x^{-2}$.

    $F(x) = -\frac{3}{x} + C$. Apply power rule: $\int 3x^{-2} dx = \frac{3x^{-1}}{-1} = -\frac{3}{x}$.

14. What is the indefinite integral of $f(x) = x^2 + 2x + 1$?

    $F(x) = \frac{x^3}{3} + x^2 + x + C$. Apply power rule to each term: perfect square trinomial.

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

    $F(x) = \text{sec}(x) + C$. Derivative of $\sec(x)$ is $\sec(x)\tan(x)$, so reverse gives this.

16. What is the indefinite integral of $f(x) = 5x^{1/2}$?

    $F(x) = \frac{10}{3}x^{3/2} + C$. Apply power rule: $\int 5x^{1/2} dx = \frac{5x^{3/2}}{3/2}$.

17. What is the indefinite integral of $f(x) = 8x^{-3}$?

    $F(x) = -4x^{-2} + C$. Apply power rule: $\int 8x^{-3} dx = \frac{8x^{-2}}{-2}$.

18. Find the indefinite integral of $f(x) = 2x + 5$.

    $F(x) = x^2 + 5x + C$. Apply power rule to each term: $\int 2x dx + \int 5 dx$.

19. What is the indefinite integral of $f(x) = 6x^5 + 3x^2$?

    $F(x) = x^6 + x^3 + C$. Apply power rule to each term separately.

20. Which rule is used to find the antiderivative of a sum $f(x) + g(x)$?

    The sum rule: $\text{F}(x) = \text{F}_1(x) + \text{F}_2(x) + C$. Antiderivative distributes over addition.

21. What is the antiderivative of $f(x) = \text{sin}(x)$?

    $F(x) = -\text{cos}(x) + C$. Derivative of $-\cos(x)$ is $\sin(x)$, so reverse gives this.

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

    $F(x) = -\text{csc}(x) + C$. Derivative of $-\csc(x)$ is $\csc(x)\cot(x)$, so reverse gives this.

23. Find the indefinite integral of $f(x) = \frac{4}{x^3}$.

    $F(x) = -2x^{-2} + C$. Rewrite as $4x^{-3}$ and apply power rule.

24. What is the antiderivative of $f(x) = \frac{1}{x}$?

    $F(x) = \text{ln}|x| + C$. Standard antiderivative of reciprocal function.

25. What is the antiderivative of $f(x) = a$ (a constant)?

    $F(x) = ax + C$. Antiderivative of constant is constant times $x$.

26. Find the indefinite integral of $f(x) = x^3 - 4x^2 + 6$.

    $F(x) = \frac{x^4}{4} - \frac{4x^3}{3} + 6x + C$. Apply power rule to each term separately.

27. State the notation for the indefinite integral of $f(x)$.

    \text{F}(x) = \text{∫} f(x) \text{dx}. Standard integral notation with differential $dx$.

28. Find the indefinite integral of $f(x) = 5x^4$.

    $F(x) = x^5 + C$. Apply power rule: $\int 5x^4 dx = \frac{5x^5}{5} = x^5$.

29. Find the indefinite integral of $f(x) = 4x^3 - 2x$.

    $F(x) = x^4 - x^2 + C$. Apply power rule to each term separately.

30. What is the indefinite integral of $f(x) = \frac{1}{x^2}$?

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

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