Finding Antiderivatives and Indefinite Integrals - AP Calculus BC
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What is the indefinite integral of $\frac{1}{x^2}$?
What is the indefinite integral of $\frac{1}{x^2}$?
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$-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.
$-\frac{1}{x} + C$. Rewrite as $x^{-2}$ and apply power rule.
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Find the antiderivative of $10e^{3x}$.
Find the antiderivative of $10e^{3x}$.
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$\frac{10}{3}e^{3x} + C$. Chain rule in reverse: multiply by reciprocal of inner derivative.
$\frac{10}{3}e^{3x} + C$. Chain rule in reverse: multiply by reciprocal of inner derivative.
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What is the integral of $\cosh x$?
What is the integral of $\cosh x$?
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$\sinh x + C$. Hyperbolic functions: sinh is antiderivative of cosh.
$\sinh x + C$. Hyperbolic functions: sinh is antiderivative of cosh.
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Find the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.
Find the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.
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$\arcsin x + C$. Standard inverse trigonometric antiderivative formula.
$\arcsin x + C$. Standard inverse trigonometric antiderivative formula.
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Evaluate $\int 4x^3 + 3x , dx$.
Evaluate $\int 4x^3 + 3x , dx$.
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$x^4 + \frac{3}{2}x^2 + C$. Apply power rule to each term: $4 \cdot \frac{x^4}{4} + 3 \cdot \frac{x^2}{2}$.
$x^4 + \frac{3}{2}x^2 + C$. Apply power rule to each term: $4 \cdot \frac{x^4}{4} + 3 \cdot \frac{x^2}{2}$.
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Identify the antiderivative of $\sinh x$.
Identify the antiderivative of $\sinh x$.
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$\cosh x + C$. Hyperbolic functions: cosh is antiderivative of sinh.
$\cosh x + C$. Hyperbolic functions: cosh is antiderivative of sinh.
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Find the indefinite integral of $5x^4$.
Find the indefinite integral of $5x^4$.
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$x^5 + C$. Apply power rule: $\int 5x^4 dx = 5 \cdot \frac{x^5}{5}$.
$x^5 + C$. Apply power rule: $\int 5x^4 dx = 5 \cdot \frac{x^5}{5}$.
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State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.
State the antiderivative of $\frac{1}{\sqrt{1-x^2}}$.
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$\arcsin x + C$. Standard inverse sine antiderivative formula.
$\arcsin x + C$. Standard inverse sine antiderivative formula.
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Evaluate $\int -\frac{1}{x} , dx$.
Evaluate $\int -\frac{1}{x} , dx$.
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$-\ln|x| + C$. Constant factor of $-1$ applied to $\ln|x|$ antiderivative.
$-\ln|x| + C$. Constant factor of $-1$ applied to $\ln|x|$ antiderivative.
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Evaluate $\int 7 , dx$.
Evaluate $\int 7 , dx$.
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$7x + C$. Constant multiplied by basic antiderivative of 1.
$7x + C$. Constant multiplied by basic antiderivative of 1.
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What is the antiderivative of $\frac{1}{1+x^2}$?
What is the antiderivative of $\frac{1}{1+x^2}$?
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$\arctan x + C$. Standard inverse tangent antiderivative formula.
$\arctan x + C$. Standard inverse tangent antiderivative formula.
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Evaluate $\int 3x^2 - 4x + 5 , dx$.
Evaluate $\int 3x^2 - 4x + 5 , dx$.
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$x^3 - 2x^2 + 5x + C$. Apply power rule to each term with appropriate coefficients.
$x^3 - 2x^2 + 5x + C$. Apply power rule to each term with appropriate coefficients.
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State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.
State the integral of $\frac{1}{\sqrt{x^2 - 1}}$.
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$\operatorname{arcsec} x + C$. Standard inverse secant antiderivative formula.
$\operatorname{arcsec} x + C$. Standard inverse secant antiderivative formula.
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Find the antiderivative of $a^x$ where $a > 0$.
Find the antiderivative of $a^x$ where $a > 0$.
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$\frac{a^x}{\ln a} + C$. General exponential antiderivative divided by natural log of base.
$\frac{a^x}{\ln a} + C$. General exponential antiderivative divided by natural log of base.
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What is the antiderivative of $\cos x$?
What is the antiderivative of $\cos x$?
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$\sin x + C$. Derivative of sine is cosine.
$\sin x + C$. Derivative of sine is cosine.
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Evaluate the indefinite integral of $2x^3$.
Evaluate the indefinite integral of $2x^3$.
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$\frac{x^4}{2} + C$. Apply power rule: $\int 2x^3 dx = 2 \cdot \frac{x^4}{4}$.
$\frac{x^4}{2} + C$. Apply power rule: $\int 2x^3 dx = 2 \cdot \frac{x^4}{4}$.
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What is the antiderivative of $\sec^2 x$?
What is the antiderivative of $\sec^2 x$?
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$\tan x + C$. Derivative of tangent function is secant squared.
$\tan x + C$. Derivative of tangent function is secant squared.
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Evaluate $\int (3x^2 + 2x) , dx$.
Evaluate $\int (3x^2 + 2x) , dx$.
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$x^3 + x^2 + C$. Apply power rule to each term separately.
$x^3 + x^2 + C$. Apply power rule to each term separately.
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Find the antiderivative of $\csc x \cot x$.
Find the antiderivative of $\csc x \cot x$.
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$-\csc x + C$. Derivative of negative cosecant is cosecant cotangent.
$-\csc x + C$. Derivative of negative cosecant is cosecant cotangent.
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What is the antiderivative of $\frac{1}{x}$?
What is the antiderivative of $\frac{1}{x}$?
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$\ln|x| + C$. Natural logarithm is the antiderivative of reciprocal.
$\ln|x| + C$. Natural logarithm is the antiderivative of reciprocal.
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State the antiderivative of $\sinh x$.
State the antiderivative of $\sinh x$.
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$\cosh x + C$. Hyperbolic sine and cosine are antiderivatives of each other.
$\cosh x + C$. Hyperbolic sine and cosine are antiderivatives of each other.
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What is the antiderivative of $x^n$ where $n \neq -1$?
What is the antiderivative of $x^n$ where $n \neq -1$?
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$\frac{x^{n+1}}{n+1} + C$. Power rule: increase exponent by 1, divide by new exponent.
$\frac{x^{n+1}}{n+1} + C$. Power rule: increase exponent by 1, divide by new exponent.
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State the antiderivative of $\tan x$.
State the antiderivative of $\tan x$.
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$ -\ln|\cos x| + C $. Standard antiderivative formula for tangent function.
$ -\ln|\cos x| + C $. Standard antiderivative formula for tangent function.
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What is the antiderivative of $\csc^2 x$?
What is the antiderivative of $\csc^2 x$?
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$ - \cot x + C $. Derivative of cotangent is negative $\csc^2 x$.
$ - \cot x + C $. Derivative of cotangent is negative $\csc^2 x$.
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What is the antiderivative of $\cosh x$?
What is the antiderivative of $\cosh x$?
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$\sinh x + C$. Hyperbolic cosine and sine are antiderivatives of each other.
$\sinh x + C$. Hyperbolic cosine and sine are antiderivatives of each other.
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Find the antiderivative of $\frac{1}{x^2 + 1}$.
Find the antiderivative of $\frac{1}{x^2 + 1}$.
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$\arctan x + C$. Standard inverse tangent antiderivative formula.
$\arctan x + C$. Standard inverse tangent antiderivative formula.
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State the antiderivative of $\sec^2 x$.
State the antiderivative of $\sec^2 x$.
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$\tan x + C$. Derivative of tangent is secant squared.
$\tan x + C$. Derivative of tangent is secant squared.
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State the antiderivative of $e^x$.
State the antiderivative of $e^x$.
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$e^x + C$. Exponential function is its own antiderivative.
$e^x + C$. Exponential function is its own antiderivative.
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Find the antiderivative of $\sin x$.
Find the antiderivative of $\sin x$.
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$-\cos x + C$. Derivative of cosine is negative sine.
$-\cos x + C$. Derivative of cosine is negative sine.
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Find the antiderivative of $\sec x \tan x$.
Find the antiderivative of $\sec x \tan x$.
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$\sec x + C$. Derivative of secant is secant tangent.
$\sec x + C$. Derivative of secant is secant tangent.
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