Exploring Accumulations of Change - AP Calculus BC

1. Calculate $\int_1^2 (2x^2 - 3) \, dx$.

   $\frac{5}{3}$. Evaluate $\frac{2x^3}{3} - 3x$ from 1 to 2.

2. What is the integral of $\csc(x) \cot(x)$?

   $-\csc(x) + C$. Standard trigonometric integration formula.

3. What does the definite integral of a positive function represent?

   The area under the curve $f(x)$ from $x = a$ to $x = b$. Geometric interpretation of definite integrals for positive functions.

4. What is the integral of $\tan(x)$?

   $-\ln|\cos(x)| + C$. Using substitution $u = \cos(x)$.

5. What is the integral of $\sin(x)$?

   $-\cos(x) + C$. Standard trigonometric integration formula.

6. What is the integral of $\csc^2(x)$?

   $-\cot(x) + C$. Standard trigonometric integration formula.

7. State the integral of $\sec^2(x)$.

   $\tan(x) + C$. Standard trigonometric integration formula.

8. What is the integral of $e^x$?

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

9. What is the integral of $\sec(x) \tan(x)$?

   $\sec(x) + C$. Standard trigonometric integration formula.

10. Evaluate: $\int_0^2 (4x^3) \, dx$.

    $16$. Evaluate $x^4$ from 0 to 2 gives $16 - 0 = 16$.

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

    $\tan(x) + C$. Since $\frac{1}{\cos^2(x)} = \sec^2(x)$.

12. What is the integral of $\cos(x)$?

    $\sin(x) + C$. Standard trigonometric integration formula.

13. What is $C$ in an antiderivative?

    A constant of integration. Represents the family of antiderivatives for indefinite integrals.

14. State the power rule for integration.

    $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, $n \neq -1$. Fundamental integration rule for polynomial functions.

15. State the integral of $\frac{1}{x^2 + 1}$.

    $\arctan(x) + C$. Standard inverse trigonometric integration formula.

16. Evaluate: $\int_0^3 (x^2) \, dx$.

    $9$. Evaluate $\frac{x^3}{3}$ from 0 to 3 gives $\frac{27}{3} = 9$.

17. Integrate: $\int 5x^4 \, dx$.

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

18. Calculate: $\int_0^1 (3x + 2) \, dx$.

    $3.5$. Evaluate $\frac{3x^2}{2} + 2x$ from 0 to 1 gives $\frac{3}{2} + 2 = 3.5$.

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

    $\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.

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

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

21. What is the substitution method in integration?

    Substitute $u = g(x)$ to simplify the integral. Technique for simplifying complex integrals by changing variables.

22. State the integration by parts formula.

    $\int u \, dv = uv - \int v \, du$. Method for integrating products of functions.

23. What is the integral of $\frac{1}{x}$?

    $\ln|x| + C$. Natural logarithm is the antiderivative of $\frac{1}{x}$.

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

    $\arctan(x) + C$. Standard inverse trigonometric integration formula.

25. Integrate: $\int (2x + 3) \, dx$.

    $x^2 + 3x + C$. Apply power rule to each term separately.

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

    $\arcsin(x) + C$. Standard inverse trigonometric integration formula.

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