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AP Calculus AB : Functions, Graphs, and Limits

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3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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Example Question #9 : Finding Definite Integrals

$\int_2^5\sqrt{x}\text{ }{\mathrm{d} x}=?$

Possible Answers:

9.34

5.56

7.52

1.89

2.59

Correct answer:

5.56

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our $f(x)=\sqrt{x}$ , we can use the power rule, if we turn it into an exponent:

$f(x)=\sqrt{x}=x^{\frac{1}{2}}$

This means that:

$\int f(x){\mathrm{d} x}=\int x^\frac{1}{2}{\mathrm{d} x}=\frac{2}{3}x^\frac{3}{2}+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\frac{2}{3}b^\frac{3}{2}+c)-(\frac{2}{3}a^\frac{3}{2}+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{2}^{5}f(x){\mathrm{d} x}=(\frac{2}{3}(5)^\frac{3}{2})-(\frac{2}{3}(2)^\frac{3}{2})$

$\int_{2}^{5}f(x){\mathrm{d} x}\approx(7.45)-(1.89)$

$\int_{2}^{5}f(x){\mathrm{d} x}\approx5.56$

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Example Question #1 : Finding Indefinite Integrals

What is the anti-derivative of ?

Possible Answers:

$\frac{1}{2}x^2+c$

x^2+c

$\frac{3}{2}x^2+c$

2+c

Correct answer:

x^2+c

Explanation:

To find the indefinite integral of our expression, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the by one and then divide by that new exponent.

First we need to realize that 2x=2x^1 . From there we can solve:

$\int2x{\mathrm{d} x}=\frac{2x^{1+1}}{1+1}+c$

When taking an integral, be sure to include a at the end of everything. stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the to anticipate the possiblity of the equation actually being x^2+5 or x^2-100 instead of just x^2 .

$\int2x{\mathrm{d} x}=\frac{2x^{2}}{2}+c$

$\int2x{\mathrm{d} x}=x^2+c$

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Example Question #8 : Finding Indefinite Integrals

What is the indefinite integral of x^3+2 ?

Possible Answers:

4x^4+2x

3x^2

$\frac{1}{3}x^4+2+c$

$\frac{1}{4}x^4+2x+c$

6x+6+c

Correct answer:

$\frac{1}{4}x^4+2x+c$

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the by one and then divide by that new exponent.

Remember that, when taking the integral, we treat constants as that number times $x^{0}$ since anything to the zero power is . For example, treat as 2x^0 .

$\int x^3+2{\mathrm{d} x}=\frac{x^{3+1}}{3+1}+\frac{2x^{0+1}}{0+1}+c$

$\int x^3+2{\mathrm{d} x}=\frac{x^{4}}{4}+\frac{2x^{1}}{1}+c$

$\int x^3+2{\mathrm{d} x}=\frac{1}{4}x^4+2x+c$

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Example Question #9 : Finding Indefinite Integrals

What is the indefinite integral of $\frac{1}{3}x^2$ ?

Possible Answers:

x^3+c

x^2

$\frac{1}{3}x^3+c$

$\frac{1}{9}x^3+c$

2x+3

Correct answer:

$\frac{1}{9}x^3+c$

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the by one and then divide by that new exponent.

$\int \frac{1}{3}x^2{\mathrm{d} x}=\frac{1}{3}*\frac{x^{2+1}}{2+1}+c$

When taking an integral, be sure to include a . stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the to anticipate the possiblity of the equation actually being x^2+5 or x^2-100 instead of just x^2 .

$\int \frac{1}{3}x^2{\mathrm{d} x}=\frac{1}{3}*\frac{x^{3}}{3}+c$

$\int \frac{1}{3}x^2{\mathrm{d} x}=\frac{1}{9}x^3+c$

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Example Question #10 : Finding Definite Integrals

$\int_{12}^{15}\ln(x){\mathrm{d} x}=?$

Possible Answers:

7.8

8.7

71.12

10.8

3.9

Correct answer:

7.8

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our $f(x)=\ln(x)$ , we can't use the power rule, as it has a special antiderivative:

$\int f(x){\mathrm{d} x}=\int \ln(x){\mathrm{d} x}=x\ln(x)-x+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(b\ln(b)-b+c)-(a\ln(a)-a+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{12}^{15}f(x){\mathrm{d} x}=(15\ln(15)-15)-(12\ln(12)-12)$

$\int_{12}^{15}f(x){\mathrm{d} x}\approx(25.62)-(17.82)$

$\int_{12}^{15}f(x){\mathrm{d} x}\approx7.8$

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Example Question #31 : Calculus Ii — Integrals

$\int_{\pi}^{2\pi}\sin(x){\mathrm{d} x}=?$

Possible Answers:

$\frac{\pi}{2}$

$2\pi$

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our $f(x)=\sin(x)$ , we can't use the power rule, as it has a special antiderivative:

$\int f(x){\mathrm{d} x}=\int \sin(x){\mathrm{d} x}=-\cos(x)+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(-\cos(b)+c)-(-\cos(a)+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(-\cos({2\pi}))-(-\cos(\pi))$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(-1)-(-(-1))$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=-2$

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Example Question #11 : Finding Definite Integrals

$\int_{\pi}^{2\pi}\cos(x){\mathrm{d} x}=?$

Possible Answers:

$\pi$

$\frac{\pi}{2}$

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our $f(x)=\cos(x)$ , we can't use the power rule, as it has a special antiderivative:

$\int f(x){\mathrm{d} x}=\int \cos(x){\mathrm{d} x}=\sin(x)+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\sin(b)+c)-(\sin(a)+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(\sin({2\pi}))-(\sin(\pi))$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(0)-(0)$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=0$

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Example Question #10 : Finding Indefinite Integrals

What is the indefinite integral of x+18 ?

Possible Answers:

Undefined

$\frac{1}{2}x^3+18x^2+cx$

$\frac{1}{2}x^2+18x+c$

2x^2+18x+c

Correct answer:

$\frac{1}{2}x^2+18x+c$

Explanation:

To find the indefinite integral of our equation, we can use the reverse power rule.

To use the reverse power rule, we raise the exponent of the by one and then divide by that new exponent.

Remember that, when taking the integral, we treat constants as that number times $x^{0}$ , since anything to the zero power is . Treat as 18x^0 .

$\int x+18{\mathrm{d} x}=\frac{x^{1+1}}{1+1}+\frac{18x^{0+1}}{0+1}+c$

$\int x+18{\mathrm{d} x}=\frac{x^{2}}{2}+\frac{18x^{1}}{1}+c$

$\int x+18{\mathrm{d} x}=\frac{1}{2}x^2+18x+c$

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Example Question #41 : Finding Integrals

What is the indefinite integral of $\sin(x)$ ?

Possible Answers:

$-\sin(x)+c$

$-\cos(x)$

$\cos(x)$

$-\cos(x)+c$

$2\sin(x^2)+c$

Correct answer:

$-\cos(x)+c$

Explanation:

$\sin(x)$ is a special function.

The indefinite integral is $\int \sin(x) {\mathrm{d} x}=-\cos(x)+c$ .

Even though it is a special function, we still need to include a . stands for "constant". Since taking the derivative of a constant whole number will always equal , we include the to anticipate the possiblity of the equation actually being x^2+5 or x^2-100 instead of just x^2 .

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Example Question #81 : Asymptotic And Unbounded Behavior

What is the indefinite integral of 6x^2+8 ?

Possible Answers:

2x^3-8x-c

2x^3+8x+c

$\frac{2}{3}x^3+8x+c$

12x

Correct answer:

2x^3+8x+c

Explanation:

To solve this problem, we can use the anti-power rule or reverse power rule. We raise the exponent on the variables by one and divide by the new exponent.

For this problem, we'll treat as 8x^0 since anything to the zero power is one.

$\int 6x^2+8{\mathrm{d} x}=\frac{6x^{2+1}}{2+1}+\frac{8x^{0+1}}{0+1}+c$

Since the derivative of any constant is , when we take the indefinite integral, we add a to compensate for any constant that might be there.

From here we can simplify.

$\int 6x^2+8{\mathrm{d} x}=\frac{6x^{3}}{3}+\frac{8x^{1}}{1}+c$

$\int 6x^2+8{\mathrm{d} x}=2x^3+8x+c$

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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #9 : Finding Definite Integrals

Example Question #1 : Finding Indefinite Integrals

Example Question #8 : Finding Indefinite Integrals

Example Question #9 : Finding Indefinite Integrals

Example Question #10 : Finding Definite Integrals

Example Question #31 : Calculus Ii — Integrals

Example Question #11 : Finding Definite Integrals

Example Question #10 : Finding Indefinite Integrals

Example Question #41 : Finding Integrals

Example Question #81 : Asymptotic And Unbounded Behavior

All AP Calculus AB Resources

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