Finding Antiderivatives and Indefinite Integrals - AP Calculus BC

1. What is the antiderivative of $\csc x \cot x$?

   $- \csc x + C$. Derivative of cosecant is negative cosecant cotangent.

2. What is the antiderivative of $\cot x$?

   $\ln|\sin x| + C$. Standard antiderivative formula for cotangent function.

3. Find the antiderivative of $\sin x$.

   $-\cos x + C$. Derivative of cosine is negative sine.

4. State the antiderivative of $\tan x$.

   $-\ln|\cos x| + C$. Standard antiderivative formula for tangent function.

5. What is the indefinite integral of $\tan x$?

   $-\ln|\cos x| + C$. Standard antiderivative formula for tangent function.

6. State the antiderivative of $e^x$.

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

7. State the antiderivative of $\sinh x$.

   $\cosh x + C$. Hyperbolic sine and cosine are antiderivatives of each other.

8. Find the indefinite integral of $5x^4$.

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

9. State the antiderivative of $\sec^2 x$.

   $\tan x + C$. Derivative of tangent is secant squared.

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

    $\ln|x| + C$. Natural logarithm is the antiderivative of reciprocal.

11. What is the integral of $\cosh x$?

    $\sinh x + C$. Hyperbolic functions: sinh is antiderivative of cosh.

12. What is the antiderivative of $\cosh x$?

    $\sinh x + C$. Hyperbolic cosine and sine are antiderivatives of each other.

13. What is the antiderivative of $x^n$ where $n \neq -1$?

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

14. Find the antiderivative of $10e^{3x}$.

    $\frac{10}{3}e^{3x} + C$. Chain rule in reverse: multiply by reciprocal of inner derivative.

15. Evaluate $\int 4x^3 + 3x \, dx$.

    $x^4 + \frac{3}{2}x^2 + C$. Apply power rule to each term: $4 \cdot \frac{x^4}{4} + 3 \cdot \frac{x^2}{2}$.

16. What is the antiderivative of $\sec^2 x$?

    $\tan x + C$. Derivative of tangent function is secant squared.

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

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

18. Identify the antiderivative of $\sinh x$.

    $\cosh x + C$. Hyperbolic functions: cosh is antiderivative of sinh.

19. Find the antiderivative of $\csc x \cot x$.

    $-\csc x + C$. Derivative of negative cosecant is cosecant cotangent.

20. What is the antiderivative of $\cos x$?

    $\sin x + C$. Derivative of sine is cosine.

21. Find the antiderivative of $\sec x \tan x$.

    $\sec x + C$. Derivative of secant is secant tangent.

22. Evaluate $\int 3x^2 - 4x + 5 \, dx$.

    $x^3 - 2x^2 + 5x + C$. Apply power rule to each term with appropriate coefficients.

23. Evaluate $\int -\frac{1}{x} \, dx$.

    $-\ln|x| + C$. Constant factor of $-1$ applied to $\ln|x|$ antiderivative.

24. Find the antiderivative of $a^x$ where $a > 0$.

    $\frac{a^x}{\ln a} + C$. General exponential antiderivative divided by natural log of base.

25. What is the antiderivative of $\csc^2 x$?

    $- \cot x + C$. Derivative of cotangent is negative $\csc^2 x$.

26. State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

27. Find the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

    $\arcsin x + C$. Standard inverse trigonometric antiderivative formula.

28. Evaluate the indefinite integral of $2x^3$.

    $\frac{x^4}{2} + C$. Apply power rule: $\int 2x^3 dx = 2 \cdot \frac{x^4}{4}$.

29. Evaluate $\int 7 \, dx$.

    $7x + C$. Constant multiplied by basic antiderivative of 1.

30. State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.

    $\operatorname{arcsec} x + C$. Standard inverse secant antiderivative formula.

31. Find the antiderivative of $\frac{1}{x^2 + 1}$.

    $\arctan x + C$. Standard inverse tangent antiderivative formula.

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

    $\arctan x + C$. Standard inverse tangent antiderivative formula.

33. Evaluate $\int (3x^2 + 2x) \, dx$.

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

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