AP Calculus BC Flashcards: Finding Antiderivatives And Indefinite Integrals

What this deck covers

This deck focuses on Finding Antiderivatives And Indefinite Integrals, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

Flashcard 2: Find the antiderivative of $10e^{3x}$.

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

Flashcard 3: What is the integral of $\cosh x$?

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

Flashcard 4: Find the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

Flashcard 5: Evaluate $\int 4x^3 + 3x \, dx$.

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

Flashcard 6: Identify the antiderivative of $\sinh x$.

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

Flashcard 7: Find the indefinite integral of $5x^4$.

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

Flashcard 8: State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

Flashcard 9: Evaluate $\int -\frac{1}{x} \, dx$.

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

Flashcard 10: Evaluate $\int 7 \, dx$.

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

Flashcard 11: What is the antiderivative of $\frac{1}{1+x^2}$?

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

Flashcard 12: Evaluate $\int 3x^2 - 4x + 5 \, dx$.

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

Flashcard 13: State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.

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

Flashcard 14: Find the antiderivative of $a^x$ where $a > 0$.

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

Flashcard 15: What is the antiderivative of $\cos x$?

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

Flashcard 16: Evaluate the indefinite integral of $2x^3$.

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

Flashcard 17: What is the antiderivative of $\sec^2 x$?

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

Flashcard 18: Evaluate $\int (3x^2 + 2x) \, dx$.

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

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

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

Flashcard 20: What is the antiderivative of $\frac{1}{x}$?

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

Flashcard 21: State the antiderivative of $\sinh x$.

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

Flashcard 22: What is the antiderivative of $x^n$ where $n \neq -1$?

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

Flashcard 23: State the antiderivative of $\tan x$.

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

Flashcard 24: What is the antiderivative of $\csc^2 x$?

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

Flashcard 25: What is the antiderivative of $\cosh x$?

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

Flashcard 26: Find the antiderivative of $\frac{1}{x^2 + 1}$.

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

Flashcard 27: State the antiderivative of $\sec^2 x$.

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

Flashcard 28: State the antiderivative of $e^x$.

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

Flashcard 29: Find the antiderivative of $\sin x$.

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

Flashcard 30: Find the antiderivative of $\sec x \tan x$.

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