Selecting Techniques for Antidifferentiation - AP Calculus BC
Card 1 of 30
Which technique is used for $xe^{x^2}$?
Which technique is used for $xe^{x^2}$?
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U-Substitution. Let $u = x^2$, then $du = 2x dx$ for substitution.
U-Substitution. Let $u = x^2$, then $du = 2x dx$ for substitution.
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What is the antiderivative of $x^n$ for $n \neq -1$?
What is the antiderivative of $x^n$ for $n \neq -1$?
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$\frac{x^{n+1}}{n+1} + C$. Power rule for integration: increase exponent by 1, divide by new exponent.
$\frac{x^{n+1}}{n+1} + C$. Power rule for integration: increase exponent by 1, divide by new exponent.
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Which method is used for $x^2 e^x$?
Which method is used for $x^2 e^x$?
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Integration by Parts. Product $x^2 e^x$ requires repeated integration by parts.
Integration by Parts. Product $x^2 e^x$ requires repeated integration by parts.
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What is the antiderivative of $e^x$?
What is the antiderivative of $e^x$?
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$e^x + C$. The derivative of $e^x$ is itself, so integration reverses this.
$e^x + C$. The derivative of $e^x$ is itself, so integration reverses this.
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What is the antiderivative of $e^{ax}$?
What is the antiderivative of $e^{ax}$?
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$\frac{1}{a}e^{ax} + C$. Chain rule in reverse: derivative of $\frac{1}{a}e^{ax}$ is $e^{ax}$.
$\frac{1}{a}e^{ax} + C$. Chain rule in reverse: derivative of $\frac{1}{a}e^{ax}$ is $e^{ax}$.
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Which method is used for $x^2 e^x$?
Which method is used for $x^2 e^x$?
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Integration by Parts. Product $x^2 e^x$ requires repeated integration by parts.
Integration by Parts. Product $x^2 e^x$ requires repeated integration by parts.
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Which technique is used for $xe^{x^2}$?
Which technique is used for $xe^{x^2}$?
Tap to reveal answer
U-Substitution. Let $u = x^2$, then $du = 2x dx$ for substitution.
U-Substitution. Let $u = x^2$, then $du = 2x dx$ for substitution.
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What is the antiderivative of $x^n$ for $n \neq -1$?
What is the antiderivative of $x^n$ for $n \neq -1$?
Tap to reveal answer
$\frac{x^{n+1}}{n+1} + C$. Power rule for integration: increase exponent by 1, divide by new exponent.
$\frac{x^{n+1}}{n+1} + C$. Power rule for integration: increase exponent by 1, divide by new exponent.
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Which technique is most appropriate for $\int \sqrt{x^2-16},dx$?
Which technique is most appropriate for $\int \sqrt{x^2-16},dx$?
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Use trig substitution: $x=4\sec(\theta)$. Square root of $x^2-a^2$ form requires secant substitution.
Use trig substitution: $x=4\sec(\theta)$. Square root of $x^2-a^2$ form requires secant substitution.
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Which technique is most appropriate for $\int x^5,dx$?
Which technique is most appropriate for $\int x^5,dx$?
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Use the power rule (basic antiderivative). Polynomial integrands use the power rule directly.
Use the power rule (basic antiderivative). Polynomial integrands use the power rule directly.
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Which technique is most appropriate for $\int e^{3x},dx$?
Which technique is most appropriate for $\int e^{3x},dx$?
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Use reverse chain rule: $\int e^{ax}dx=\frac{1}{a}e^{ax}+C$. The coefficient $a$ in $e^{ax}$ moves to the denominator.
Use reverse chain rule: $\int e^{ax}dx=\frac{1}{a}e^{ax}+C$. The coefficient $a$ in $e^{ax}$ moves to the denominator.
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Which technique is most appropriate for $\int \cos(7x),dx$?
Which technique is most appropriate for $\int \cos(7x),dx$?
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Use reverse chain rule: $\int \cos(ax)dx=\frac{1}{a}\sin(ax)+C$. The coefficient $a$ in $\cos(ax)$ moves to the denominator.
Use reverse chain rule: $\int \cos(ax)dx=\frac{1}{a}\sin(ax)+C$. The coefficient $a$ in $\cos(ax)$ moves to the denominator.
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Which technique is most appropriate for $\int \frac{1}{x},dx$?
Which technique is most appropriate for $\int \frac{1}{x},dx$?
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Use the logarithm rule: $\int \frac{1}{x}dx=\ln|x|+C$. The integral of $\frac{1}{x}$ is the natural logarithm.
Use the logarithm rule: $\int \frac{1}{x}dx=\ln|x|+C$. The integral of $\frac{1}{x}$ is the natural logarithm.
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Which technique is most appropriate for $\int \frac{1}{2x-5},dx$?
Which technique is most appropriate for $\int \frac{1}{2x-5},dx$?
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Use reverse chain rule: $\frac{1}{2}\ln|2x-5|+C$. Linear denominator requires chain rule adjustment by $\frac{1}{2}$.
Use reverse chain rule: $\frac{1}{2}\ln|2x-5|+C$. Linear denominator requires chain rule adjustment by $\frac{1}{2}$.
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Which technique is most appropriate for $\int (x^2+1)^7\cdot 2x,dx$?
Which technique is most appropriate for $\int (x^2+1)^7\cdot 2x,dx$?
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Use $u$-substitution with $u=x^2+1$. The derivative of $u=x^2+1$ is $2x$, which appears in the integrand.
Use $u$-substitution with $u=x^2+1$. The derivative of $u=x^2+1$ is $2x$, which appears in the integrand.
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Which technique is most appropriate for $\int \frac{x}{x^2+9},dx$?
Which technique is most appropriate for $\int \frac{x}{x^2+9},dx$?
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Use $u$-substitution with $u=x^2+9$. The numerator $x$ is half the derivative of the denominator.
Use $u$-substitution with $u=x^2+9$. The numerator $x$ is half the derivative of the denominator.
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Which technique is most appropriate for $\int x\sqrt{x^2+4},dx$?
Which technique is most appropriate for $\int x\sqrt{x^2+4},dx$?
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Use $u$-substitution with $u=x^2+4$. The $x$ factor is half the derivative of what's under the radical.
Use $u$-substitution with $u=x^2+4$. The $x$ factor is half the derivative of what's under the radical.
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Which technique is most appropriate for $\int x e^{x^2},dx$?
Which technique is most appropriate for $\int x e^{x^2},dx$?
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Use $u$-substitution with $u=x^2$. The $x$ factor is half the derivative of the exponent.
Use $u$-substitution with $u=x^2$. The $x$ factor is half the derivative of the exponent.
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Which technique is most appropriate for $\int x\ln(x),dx$?
Which technique is most appropriate for $\int x\ln(x),dx$?
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Use integration by parts. Products of polynomials and logarithms require parts.
Use integration by parts. Products of polynomials and logarithms require parts.
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Which technique is most appropriate for $\int x\sin(x),dx$?
Which technique is most appropriate for $\int x\sin(x),dx$?
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Use integration by parts. Products of polynomials and trig functions require parts.
Use integration by parts. Products of polynomials and trig functions require parts.
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Which technique is most appropriate for $\int e^x\cos(x),dx$?
Which technique is most appropriate for $\int e^x\cos(x),dx$?
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Use integration by parts twice (tabular/recursive). Products of exponentials and trig functions need repeated parts.
Use integration by parts twice (tabular/recursive). Products of exponentials and trig functions need repeated parts.
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Which technique is most appropriate for $\int \frac{x^2+1}{x-3},dx$?
Which technique is most appropriate for $\int \frac{x^2+1}{x-3},dx$?
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Use algebraic division, then integrate term-by-term. Improper rational function requires polynomial division first.
Use algebraic division, then integrate term-by-term. Improper rational function requires polynomial division first.
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Which technique is most appropriate for $\int \frac{1}{x^2-9},dx$?
Which technique is most appropriate for $\int \frac{1}{x^2-9},dx$?
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Use partial fractions (factor $x^2-9=(x-3)(x+3)$). Factorable quadratic denominator uses partial fractions.
Use partial fractions (factor $x^2-9=(x-3)(x+3)$). Factorable quadratic denominator uses partial fractions.
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Which technique is most appropriate for $\int \frac{1}{x^2+9},dx$?
Which technique is most appropriate for $\int \frac{1}{x^2+9},dx$?
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Use arctangent form: $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$. Matches the arctangent integral form with $a=3$.
Use arctangent form: $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan(\frac{x}{a})+C$. Matches the arctangent integral form with $a=3$.
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Which technique is most appropriate for $\int \frac{1}{\sqrt{9-x^2}},dx$?
Which technique is most appropriate for $\int \frac{1}{\sqrt{9-x^2}},dx$?
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Use arcsine form: $\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin(\frac{x}{a})+C$. Matches the arcsine integral form with $a=3$.
Use arcsine form: $\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin(\frac{x}{a})+C$. Matches the arcsine integral form with $a=3$.
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Which technique is most appropriate for $\int \sqrt{9-x^2},dx$?
Which technique is most appropriate for $\int \sqrt{9-x^2},dx$?
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Use trig substitution: $x=3\sin(\theta)$. Square root of $a^2-x^2$ form requires sine substitution.
Use trig substitution: $x=3\sin(\theta)$. Square root of $a^2-x^2$ form requires sine substitution.
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Which technique is most appropriate for $\int \sqrt{x^2+16},dx$?
Which technique is most appropriate for $\int \sqrt{x^2+16},dx$?
Tap to reveal answer
Use trig substitution: $x=4\tan(\theta)$. Square root of $x^2+a^2$ form requires tangent substitution.
Use trig substitution: $x=4\tan(\theta)$. Square root of $x^2+a^2$ form requires tangent substitution.
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Which technique is most appropriate for $\int \sin^3(x)\cos^2(x),dx$?
Which technique is most appropriate for $\int \sin^3(x)\cos^2(x),dx$?
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Use trig identities; save a $\sin(x)$ and substitute $u=\cos(x)$. Odd power of sine allows $u$-substitution after saving one sine.
Use trig identities; save a $\sin(x)$ and substitute $u=\cos(x)$. Odd power of sine allows $u$-substitution after saving one sine.
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What is the antiderivative of $e^{ax}$?
What is the antiderivative of $e^{ax}$?
Tap to reveal answer
$\frac{1}{a}e^{ax} + C$. Chain rule in reverse: derivative of $\frac{1}{a}e^{ax}$ is $e^{ax}$.
$\frac{1}{a}e^{ax} + C$. Chain rule in reverse: derivative of $\frac{1}{a}e^{ax}$ is $e^{ax}$.
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What is the antiderivative of $e^x$?
What is the antiderivative of $e^x$?
Tap to reveal answer
$e^x + C$. The derivative of $e^x$ is itself, so integration reverses this.
$e^x + C$. The derivative of $e^x$ is itself, so integration reverses this.
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