Applying Properties of Definite Integrals - AP Calculus AB

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

   $\text{sin}(x) + C$. Standard antiderivative: derivative of sine is cosine.

2. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.

   $\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.

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

   $-\text{cos}(x) + C$. Standard antiderivative: derivative of negative cosine is sine.

4. What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

   $-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.

5. What is the integral of $x^n$ with respect to $x$?

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

6. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.

   $\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.

7. Find the derivative of $e^{2x}$.

   $2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.

8. Evaluate $\frac{d}{dx}\bigg(\frac{1}{x^2 + 1}\bigg)$.

   $-\frac{2x}{(x^2 + 1)^2}$. Chain rule on $(x^2 + 1)^{-1}$ gives $-1 \cdot (x^2 + 1)^{-2} \cdot 2x$.

9. State the integral of $\frac{1}{x}$ with respect to $x$.

   $\text{ln}|x| + C$. Standard antiderivative formula for $\frac{1}{x}$.

10. What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?

    $-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.

11. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.

    $\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.

12. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\tan(x)}\bigg)$.

    $-\frac{1}{\sin^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.

13. What is the integral of $\text{e}^{ax}$?

    $\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.

14. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.

    $-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.

15. Evaluate $\frac{d}{dx} \bigg(\frac{1}{x}\bigg)$.

    $-\frac{1}{x^2}$. Power rule: $\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2}$.

16. Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.

    $\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.

17. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{tan}(x)}\bigg)$.

    $-\frac{1}{\text{sin}^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.

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

    $e^x + C$. The exponential function is its own derivative and antiderivative.

19. Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.

    $-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.

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