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AP Calculus AB : Functions, Graphs, and Limits

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Example Questions

Example Question #1 : Finding Integrals By Substitution

Evaluate:

$\dpi{120}\int_{1}^{e^{ 3}}\frac{\sqrt{\ln x}}{x}\; \; dx$

Possible Answers:

$\frac{2}{3}$

$2\sqrt{3}$

$2 \ln 3$

$\frac{2\sqrt{3}}{3}$

$2 \sqrt{\ln 3}$

Correct answer:

$2\sqrt{3}$

Explanation:

Set $u = \ln x$ . Then

$\frac{\mathrm{d} u}{\mathrm{d} x} = \frac{\mathrm{d} }{\mathrm{d} x} \ln x = \frac{1}{x}$

and

$du = \frac{dx}{x}$

Also, since $u = \ln x$ , the limits of integration change to $\ln e^{3} = 3$ and $\ln 1= 0$ .

Substitute:

$\dpi{120}\int_{1}^{e^{ 3}}\frac{\sqrt{\ln x}}{x}\; \; dx$

$\dpi{120} = \int_{0}^{3}\sqrt{u} \; du$

$= \int_{0}^{3} u ^{\frac{1}{2}} \; du$

$\dpi{150} = \left.\begin{matrix} \frac{u^{\frac{3}{2}}}{{\frac{3}{2}}} & \\ & \end{matrix}\right|$ $\begin{matrix} 3\\ \\ 0 \end{matrix}$

$\dpi{150} = \left.\begin{matrix} \\ \frac{ 2u \sqrt{u}} {3} & \\ & \end{matrix}\right|$ $\begin{matrix} 3\\ \\ 0 \end{matrix}$

$=\frac{ 2\cdot 3\cdot \sqrt{3}} {3} - \frac{ 2\cdot 0\cdot \sqrt{0}} {3} =2\sqrt{3}$

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Example Question #2 : Finding Definite Integrals

$\int_{2}^{4}x^2{\mathrm{d} x}=?$

Possible Answers:

$28\tfrac{2}{3}$

$5\tfrac{1}{2}$

$18\frac{2}{3}$

$22\tfrac{1}{3}$

$11\tfrac{3}{4}$

Correct answer:

$18\frac{2}{3}$

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)=x^2 , we can use the reverse power rule to find the indefinite integral or anti-derivative of our function:

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

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

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\frac{1}{3}b^3+c)-(\frac{1}{3}a^3+c)$

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

$\int_{2}^{4}f(x){\mathrm{d} x}=(\frac{1}{3}\cdot 4^3)-(\frac{1}{3}\cdot 8^3)$

$\int_{2}^{4}f(x){\mathrm{d} x}=(\frac{64}{3})-(\frac{8}{3})$

$\int_{2}^{4}f(x){\mathrm{d} x}=\frac{56}{3}=18\frac{2}{3}$

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Example Question #73 : Functions, Graphs, And Limits

$\int_{1}^{2}\frac{1}{x}{\mathrm{d} x}=?$

Possible Answers:

0.81

0.75

1.2

0.30

0.69

Correct answer:

0.69

Explanation:

Remember the fundamental theorem of calculus!

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

As it turns out, since our $f(x)=\frac{1}{x}$ , the power rule really doesn't help us. $\frac{1}{x}$ has a special anti derivative: $\ln{x}$ .

$\int \frac{1}{x}{\mathrm{d} x}=\ln{x}+c$

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

Now we can plug that equation into our FToC equation:

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

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

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

$\int_{1}^{2}f(x){\mathrm{d} x}=(\ln(2))-(0)$

$\int_{1}^{2}f(x){\mathrm{d} x}\approx 0.69$

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Example Question #3 : Finding Definite Integrals

$\int_{3}^{40}e^x{\mathrm{d} x}=?$

Possible Answers:

$2.35\cdot 10^{9}$

$2.35\cdot 10^{19}$

$1.05\cdot 10^{17}$

$2.35\cdot 10^{15}$

$2.35\cdot 10^{17}$

Correct answer:

$2.35\cdot 10^{17}$

Explanation:

Remember the fundamental theorem of calculus!

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

As it turns out, since our f(x)=e^x , the power rule really doesn't help us. e^x is the only function that is it's OWN anti-derivative. That means we're still going to be working with e^x .

$\int e^x{\mathrm{d} x}=e^x+c$

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

Now we can plug that equation into our FToC equation:

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

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

$\int_{3}^{40}f(x){\mathrm{d} x}=(e^{40})-(e^3)$

$\int_{3}^{40}f(x){\mathrm{d} x}=(2.35\cdot 10^{17})-(20.09)$

Because e^3 is so small in comparison to the value we got for $e^{40}$ , our answer will end up being $2.35\cdot 10^{17}$

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Example Question #34 : Integrals

What is the indefinite integral of x^2+5x ?

Possible Answers:

10x^2+c

3x+5

x^4+5x^2+c

$\frac{1}{3}x^3+\frac{5}{2}x^2+c$

2x+5

Correct answer:

$\frac{1}{3}x^3+\frac{5}{2}x^2+c$

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

$\int x^2+5x{\mathrm{d} x}=\frac{x^{2+1}}{2+1}+\frac{5x^{1+1}}{1+1}+c$

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

$\int x^2+5x{\mathrm{d} x}=\frac{x^{3}}{3}+\frac{5x^{2}}{2}+c$

$\int x^2+5x{\mathrm{d} x}=\frac{1}{3}x^3+\frac{5}{2}x^2+c$

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Example Question #73 : Functions, Graphs, And Limits

What is the indefinite integral of 5x+8 ?

Possible Answers:

$\frac{5}{2}x^2+8x +c$

16x^3+c

$\frac{5}{6}x^3+4x^2+cx+b$

Correct answer:

$\frac{5}{2}x^2+8x +c$

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent. For this problem, that would look like:

$\int 5x+8{\mathrm{d} x}=\frac{5x^{1+1}}{1+1}+\frac{8x^{0+1}}{0+1}+c$

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

$\int 5x+8{\mathrm{d} x}=\frac{5x^{2}}{2}+\frac{8x^{1}}{1}+c$

$\int 5x+8{\mathrm{d} x}=\frac{5}{2}x^2+8x +c$

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Example Question #31 : Integrals

What is the indefinite integral of x^3+2x+5 ?

Possible Answers:

4x^4+x^2+5x+c

$\frac{1}{4}x^4+2x^2+5x+c$

$\frac{1}{4}x^4+x^2+5x+c$

3x^2+2

Correct answer:

$\frac{1}{4}x^4+x^2+5x+c$

Explanation:

To solve for the indefinite integral, we can use the reverse power rule. We raise the power of the exponents by one and divide by that new exponent.

We're going to treat as 5x^0 , as anything to the zero power is one.

For this problem, that would look like:

$\int x^3+2x+5{\mathrm{d} x}=\frac{x^{3+1}}{3+1}+\frac{2x^{1+1}}{1+1}+\frac{5x^{0+1}}{0+1}+c$

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

$\int x^3+2x+5{\mathrm{d} x}=\frac{x^{4}}{4}+\frac{2x^{2}}{2}+\frac{5x^{1}}{1}+c$

$\int x^3+2x+5{\mathrm{d} x}=\frac{1}{4}x^4+x^2+5x+c$

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Example Question #71 : Functions, Graphs, And Limits

Determine the indefinite integral:

$\dpi{120} \int\cos x \sqrt{\sin x }\; \; dx$

Possible Answers:

$\dpi{120} \frac{ 2 \sqrt{\sin x }} {3} +C$

$\dpi{120} \dpi{120} \frac{ 2 \cos x \cdot \sqrt{\cos x }} {3} +C$

$\frac{ 2 \sin x \cdot \sqrt{\sin x }} {3} +C$

$\dpi{120} \dpi{120} \frac{ 2 \sqrt{\cos x }} {3} +C$

$\dpi{120} \frac{ 2 \sin x \cdot \sqrt{\cos x }} {3} +C$

Correct answer:

$\frac{ 2 \sin x \cdot \sqrt{\sin x }} {3} +C$

Explanation:

Set $u = \sin x$ . Then

$\frac{\mathrm{du} }{\mathrm{d} x}= \frac{\mathrm{d} }{\mathrm{d} x} \sin x= \cos x$ .

and

$\cos x \; dx = du$

The integral becomes:

$\dpi{120} \int\cos x \sqrt{\sin x }\; \; dx$

$\dpi{120} =\int \sqrt{u }\; \; du$

$= \int u ^{\frac{1}{2}} \; du$

$\dpi{120} \dpi{150} = \frac{u^{\frac{1}{2}+1}}{{\frac{1}{2}+1}} + C$

$\dpi{150} = \frac{u^{\frac{3}{2}}}{{\frac{3}{2}}} + C$

$\dpi{120} =\frac{ 2u \sqrt{u}} {3} +C$

Substitute back:

$\dpi{120} =\frac{ 2 \sin x \cdot \sqrt{\sin x }} {3} +C$

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Example Question #1 : Finding Definite Integrals

$\int_2^6(5x^2+3x+2){\mathrm{d} x}=?$

Possible Answers:

$26\tfrac{2}{3}$

$401\tfrac{1}{3}$

$204\tfrac{2}{3}$

$-12\tfrac{2}{3}$

$402\tfrac{2}{3}$

Correct answer:

$402\tfrac{2}{3}$

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)=5x^2+3x+2 , we can use the power rule for all of the terms involved to find our anti-derivative:

$\int (5x^2+3x+2){\mathrm{d} x}=\frac{5}{3}x^3+\frac{3}{2}x^2+2x+c$

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

Now we can plug that equation into our FToC equation:

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

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

$\int_{2}^{6}f(x){\mathrm{d} x}=(\frac{5}{3}(6)^3+\frac{3}{2}(6)^2+2(6))-(\frac{5}{3}(2)^3+\frac{3}{2}(2)^2+2(2))$

$\int_{2}^{6}f(x){\mathrm{d} x}=(360+54+12)-(\frac{40}{3}+6+4)$

$\int_{2}^{6}f(x){\mathrm{d} x}=(426)-(23\tfrac{1}{3})$

$\int_{2}^{6}f(x){\mathrm{d} x}=402\tfrac{2}{3}$

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Example Question #1 : Finding Integrals

$\int_3^5\frac{x+3}{x}{\mathrm{d} x}=?$

Possible Answers:

3.21

1.33

5.67

6.12

3.53

Correct answer:

3.53

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 $f(x)=\frac{x+3}{x}$ , we can't use the power rule. We have to break up the quotient into separate parts:

$f(x)=\frac{x}{x}+\frac{3}{x}=1+\frac{3}{x}$ .

The integral of 1 should be no problem, but the other half is a bit more tricky:

$\int\frac{3}{x}{\mathrm{d} x}$ is really the same as $3\int\frac{1}{x}{\mathrm{d} x}$ . Since $\int\frac{1}{x}{\mathrm{d} x}=\ln{x}$ , $\int\frac{3}{x}{\mathrm{d} x}=3\ln{x}$ .

Therefore:

$\int\frac{x+3}{x}{\mathrm{d} x}=x+3\ln{x}+c$

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

Now we can plug that equation into our FToC equation:

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

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

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

$\int_{3}^{5}f(x){\mathrm{d} x}=(9.83)-(6.30)$

$\int_{3}^{5}f(x){\mathrm{d} x}\approx 3.53$

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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 #1 : Finding Integrals By Substitution

Example Question #2 : Finding Definite Integrals

Example Question #73 : Functions, Graphs, And Limits

Example Question #3 : Finding Definite Integrals

Example Question #34 : Integrals

Example Question #73 : Functions, Graphs, And Limits

Example Question #31 : Integrals

Example Question #71 : Functions, Graphs, And Limits

Example Question #1 : Finding Definite Integrals

Example Question #1 : Finding Integrals

All AP Calculus AB Resources

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