Applying Properties of Definite Integrals - AP Calculus AB
Card 1 of 30
What is the antiderivative of $e^x$?
What is the antiderivative of $e^x$?
Tap to reveal answer
$e^x + C$. The exponential function is its own derivative and antiderivative.
$e^x + C$. The exponential function is its own derivative and antiderivative.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.
Tap to reveal answer
$\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.
$\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.
Tap to reveal answer
$\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.
$\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.
← Didn't Know|Knew It →
What is the antiderivative of $\text{sin}(x)$?
What is the antiderivative of $\text{sin}(x)$?
Tap to reveal answer
$-\text{cos}(x) + C$. Standard antiderivative: derivative of negative cosine is sine.
$-\text{cos}(x) + C$. Standard antiderivative: derivative of negative cosine is sine.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.
Tap to reveal answer
$-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.
$-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.
Tap to reveal answer
$\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.
$\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx} \bigg(\frac{1}{x}\bigg)$.
Evaluate $\frac{d}{dx} \bigg(\frac{1}{x}\bigg)$.
Tap to reveal answer
$-\frac{1}{x^2}$. Power rule: $\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2}$.
$-\frac{1}{x^2}$. Power rule: $\frac{d}{dx}(x^{-1}) = -1 \cdot x^{-2}$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{x^2 + 1}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{x^2 + 1}\bigg)$.
Tap to reveal answer
$-\frac{2x}{(x^2 + 1)^2}$. Chain rule on $(x^2 + 1)^{-1}$ gives $-1 \cdot (x^2 + 1)^{-2} \cdot 2x$.
$-\frac{2x}{(x^2 + 1)^2}$. Chain rule on $(x^2 + 1)^{-1}$ gives $-1 \cdot (x^2 + 1)^{-2} \cdot 2x$.
← Didn't Know|Knew It →
What is the integral of $\text{cos}(x)$?
What is the integral of $\text{cos}(x)$?
Tap to reveal answer
$\text{sin}(x) + C$. Standard antiderivative: derivative of sine is cosine.
$\text{sin}(x) + C$. Standard antiderivative: derivative of sine is cosine.
← Didn't Know|Knew It →
Find the derivative of $e^{2x}$.
Find the derivative of $e^{2x}$.
Tap to reveal answer
$2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.
$2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.
Tap to reveal answer
$-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.
$-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{tan}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{tan}(x)}\bigg)$.
Tap to reveal answer
$-\frac{1}{\text{sin}^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.
$-\frac{1}{\text{sin}^2(x)}$. Chain rule on $\cot(x) = (\tan(x))^{-1}$ gives $-\csc^2(x)$.
← Didn't Know|Knew It →
What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
Tap to reveal answer
$-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.
$-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.
← Didn't Know|Knew It →
Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.
Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.
Tap to reveal answer
$\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.
$\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.
← Didn't Know|Knew It →
State the integral of $\frac{1}{x}$ with respect to $x$.
State the integral of $\frac{1}{x}$ with respect to $x$.
Tap to reveal answer
$\text{ln}|x| + C$. Standard antiderivative formula for $\frac{1}{x}$.
$\text{ln}|x| + C$. Standard antiderivative formula for $\frac{1}{x}$.
← Didn't Know|Knew It →
What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
Tap to reveal answer
$-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.
$-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.
← Didn't Know|Knew It →
What is the integral of $x^n$ with respect to $x$?
What is the integral of $x^n$ with respect to $x$?
Tap to reveal answer
$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.
$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.
← Didn't Know|Knew It →
What is the integral of $\text{e}^{ax}$?
What is the integral of $\text{e}^{ax}$?
Tap to reveal answer
$\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.
$\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.
← Didn't Know|Knew It →
State the integral of $\frac{1}{x}$ with respect to $x$.
State the integral of $\frac{1}{x}$ with respect to $x$.
Tap to reveal answer
$\ln |x| + C$. Standard antiderivative formula for $\frac{1}{x}$.
$\ln |x| + C$. Standard antiderivative formula for $\frac{1}{x}$.
← Didn't Know|Knew It →
What is the integral of $\text{e}^{ax}$?
What is the integral of $\text{e}^{ax}$?
Tap to reveal answer
$\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.
$\frac{1}{a}\text{e}^{ax} + C$. Standard exponential integral with constant $a$.
← Didn't Know|Knew It →
What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
What is the value of $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
Tap to reveal answer
$-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.
$-\frac{1}{x^2}$. Power rule applied to $x^{-1}$.
← Didn't Know|Knew It →
Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.
Find $\frac{d}{dx} \bigg( \text{ln}|x| \bigg)$.
Tap to reveal answer
$\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.
$\frac{1}{x}$. Derivative of natural logarithm is $\frac{1}{x}$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{sin}(x)}\bigg)$.
Tap to reveal answer
$-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.
$-\frac{\text{cos}(x)}{\text{sin}^2(x)}$. Chain rule on $\csc(x) = (\sin(x))^{-1}$ gives $-\csc(x)\cot(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cot}(x)}\bigg)$.
Tap to reveal answer
$\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.
$\frac{1}{\text{sin}^2(x)}$. Chain rule on $\tan(x) = (\cot(x))^{-1}$ gives $\sec^2(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{csc}(x)}\bigg)$.
Tap to reveal answer
$\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.
$\text{cot}(x)\text{csc}(x)$. Chain rule on $\sin(x) = (\csc(x))^{-1}$ gives $\cos(x)$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{cos}(x)}\bigg)$.
Tap to reveal answer
$\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.
$\frac{\text{sin}(x)}{\text{cos}^2(x)}$. Chain rule on $\sec(x) = (\cos(x))^{-1}$ gives $\sec(x)\tan(x)$.
← Didn't Know|Knew It →
Find the derivative of $e^{2x}$.
Find the derivative of $e^{2x}$.
Tap to reveal answer
$2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.
$2e^{2x}$. Chain rule: $\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2$.
← Didn't Know|Knew It →
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.
Evaluate $\frac{d}{dx}\bigg(\frac{1}{\text{e}^{ax}}\bigg)$.
Tap to reveal answer
$-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.
$-a\text{e}^{-ax}$. Chain rule on $e^{-ax}$ gives $-a \cdot e^{-ax}$.
← Didn't Know|Knew It →
What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
What is $\frac{d}{dx}\bigg(\frac{1}{x}\bigg)$?
Tap to reveal answer
$-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.
$-\frac{1}{x^2}$. Derivative of $x^{-1}$ using power rule.
← Didn't Know|Knew It →
What is the integral of $x^n$ with respect to $x$?
What is the integral of $x^n$ with respect to $x$?
Tap to reveal answer
$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.
$\frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Power rule for integration increases exponent by 1.
← Didn't Know|Knew It →