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 antiderivative of $\csc x \cot x$?

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

Flashcard 2: 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 3: What is the antiderivative of $\cot x$?

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

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

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

Flashcard 5: What is the antiderivative of $\csc x \cot x$?

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

Flashcard 6: Find the antiderivative of $\sin x$.

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

Flashcard 7: What is the antiderivative of $\frac{1}{x}$?

Answer: $\ln|x| + C$. Natural log is antiderivative of reciprocal function.

Flashcard 8: State the antiderivative of $e^x$.

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

Flashcard 9: State the antiderivative of $\sinh x$.

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

Flashcard 10: State the antiderivative of $\tan x$.

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

Flashcard 11: What is the indefinite integral of $\tan x$?

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

Flashcard 12: State the antiderivative of $e^x$.

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

Flashcard 13: State the antiderivative of $\sinh x$.

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

Flashcard 14: State the antiderivative of $\tan x$.

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

Flashcard 15: 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 16: Find the antiderivative of $\sec x \tan x$.

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

Flashcard 17: 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 18: What is the antiderivative of $\frac{1}{1+x^2}$?

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

Flashcard 19: State the antiderivative of $\sec^2 x$.

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

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

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

Flashcard 21: What is the antiderivative of $\frac{1}{x}$?

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

Flashcard 22: What is the integral of $\cosh x$?

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

Flashcard 23: What is the antiderivative of $\cosh x$?

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

Flashcard 24: 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 25: Evaluate $\int (3x^2 + 2x) \, dx$.

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

Flashcard 26: State the antiderivative of $\sec^2 x$.

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

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

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

Flashcard 28: What is the antiderivative of $\cosh x$?

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

Flashcard 29: What is the antiderivative of $\sec^2 x$?

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

Flashcard 30: Find the antiderivative of $10e^{3x}$.

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

Flashcard 31: 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 32: What is the antiderivative of $\sec^2 x$?

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

Flashcard 33: What is the indefinite integral of $\frac{1}{x^2}$?

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

Flashcard 34: What is the indefinite integral of $\frac{1}{x^2}$?

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

Flashcard 35: Find the antiderivative of $\frac{1}{x^2 + 1}$.

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

Flashcard 36: Identify the antiderivative of $\sinh x$.

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

Flashcard 37: Identify the antiderivative of $\sinh x$.

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

Flashcard 38: Find the antiderivative of $\csc x \cot x$.

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

Flashcard 39: Evaluate $\int -\frac{1}{x} \, dx$.

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

Flashcard 40: What is the antiderivative of $\cos x$?

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

Flashcard 41: Find the antiderivative of $\sec x \tan x$.

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

Flashcard 42: 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 43: Evaluate $\int 3x^2 - 4x + 5 \, dx$.

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

Flashcard 44: What is the indefinite integral of $\tan x$?

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

Flashcard 45: State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.

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

Flashcard 46: Evaluate $\int -\frac{1}{x} \, dx$.

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

Flashcard 47: 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 48: What is the antiderivative of $\csc^2 x$?

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

Flashcard 49: State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

Flashcard 50: Find the antiderivative of $\sin x$.

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

Flashcard 51: State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

Flashcard 52: Evaluate $\int 7 \, dx$.

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

Flashcard 53: What is the antiderivative of $\csc^2 x$?

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

Flashcard 54: Find the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.

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

Flashcard 55: What is the integral of $\cosh x$?

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

Flashcard 56: 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 57: What is the antiderivative of $\cos x$?

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

Flashcard 58: 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 59: Evaluate $\int 7 \, dx$.

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

Flashcard 60: Find the antiderivative of $10e^{3x}$.

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

Flashcard 61: State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.

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

Flashcard 62: 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 63: Find the antiderivative of $\frac{1}{x^2 + 1}$.

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

Flashcard 64: What is the antiderivative of $\frac{1}{1+x^2}$?

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

Flashcard 65: What is the antiderivative of $\cot x$?

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

Flashcard 66: Evaluate $\int (3x^2 + 2x) \, dx$.

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