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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

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

Example Questions

Example Question #23 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate:

$\int (x^2 +\frac{1}{2x})dx$

Possible Answers:

$\frac{x^3}{3}+\frac{1}{2}\ln\left | x\right |$

$x^3+\frac{1}{2}\ln\left | x\right |+C$

$\frac{x^3}{3}+\frac{1}{2}\ln\left | x\right |+C$

$\frac{x^3}{3}+\frac{2}{\ln(x)}+C$

Correct answer:

$\frac{x^3}{3}+\frac{1}{2}\ln\left | x\right |+C$

Explanation:

To evaluate the integral, we can split it into two integrals:

$\int x^2 dx+ \frac{1}{2}\int \frac{dx}{x}$

After integrating, we get

$\frac{x^3}{3}+\frac{1}{2}\ln\left | x\right |+C$

where a single constant of integration comes from the sum of the two integration constants from the two individual integrals, added together.

The rules used to integrate are

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$ , $\int \frac{dx}{x}=\ln\left | x\right | +C$

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Example Question #24 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Solve:

$\int \frac{dx}{12+9x^2}$

Possible Answers:

$\frac{1}{4\sqrt{3}}\arctan(\frac{3x}{\sqrt{3}})+C$

$\frac{1}{2\sqrt{3}}\arctan(\frac{x}{2\sqrt{3}})+C$

$\arctan(\frac{3x}{2\sqrt{3}})+C$

$\frac{1}{2\sqrt{3}}\arctan(\frac{3x}{2\sqrt{3}})+C$

Correct answer:

$\frac{1}{2\sqrt{3}}\arctan(\frac{3x}{2\sqrt{3}})+C$

Explanation:

The integral is equal to

$\frac{1}{2\sqrt{3}}\arctan(\frac{3x}{2\sqrt{3}})+C$

and was found using the following rule:

$\int \frac{dx}{a^2+x^2}=\frac{1}{a}\arctan{\frac{x}{a}}+C$

where $a=\sqrt{12}=2\sqrt{3}$

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Example Question #25 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Solve:

$\int (\frac{1}{x^2}+\frac{1}{3x})dx$

Possible Answers:

$\frac{1}{x}+\frac{1}{3}\ln \left | x\right |+C$

$-\frac{1}{x}+\frac{1}{3}\ln \left | x\right |$

$-\frac{1}{x}+\frac{1}{3}\ln \left | x\right |+C$

$-\frac{1}{x}+\frac{1}{3}\ln \left | 3x\right |+C$

Correct answer:

$-\frac{1}{x}+\frac{1}{3}\ln \left | x\right |+C$

Explanation:

To integrate, we can split the integral into the sum of two separate integrals:

$\int \frac{dx}{x^2}+\frac{1}{3}\int \frac{dx}{x}$

Integrating, we get

$-\frac{1}{x}+\frac{1}{3}\ln\left | x\right |+C$

which was found using the following rules:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$ , $\int \frac{dx}{x}=\ln\left | x\right |+C$

Note that the constants of integration were combined to make a single integration constant in the final answer.

(The first integral can be rewritten as $\int x^{-2}dx$ for clarity.)

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Example Question #31 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Calculate the integral in the following expression:

$\int(e^2)^{\ln 3x}dx$

Possible Answers:

x^3 + C

3x^3 + C

$\frac{(e^2)^{\ln3x}}{2x} + C$

$6x(e^2)^{\ln 3x} + C$

$\frac{2^{3x}}{\ln(2)} + C$

Correct answer:

3x^3 + C

Explanation:

Solving this integral depends on knowledge of exponent rules; mainly, that $(a^b)^c = a^{b*c}$ . Using this, we can simplify the given expression.

$\int(e^2)^{\ln3x}dx = \int e^{2 * \ln 3x}dx = \int (e^{\ln 3x})^2dx =\int (3x)^2dx =\int 9x^2dx$

From here, we just follow the power rule, raising the exponent and dividing by it.

$9\int x^2 dx = 3x^3 + C$

Giving us our final answer.

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Example Question #101 : Integrals

Evaluate the integral

$\int (4x^4+2x^2+3x )dx$

Possible Answers:

4x^5+2x^3+3x^2

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

$\frac{4x^5}{5}+\frac{2x^3}{3}+\frac{3x^2}{2}+C$

$\frac{x^4}{5}+\frac{x^2}{3}+\frac{x^2}{2}+C$

Correct answer:

$\frac{4x^5}{5}+\frac{2x^3}{3}+\frac{3x^2}{2}+C$

Explanation:

To evaluate the integral, we use the following definition

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$

$\int (4x^4+2x^2+3x )dx=\frac{4x^5}{5}+\frac{2x^3}{3}+\frac{3x^2}{2}+C$

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Example Question #34 : Techniques Of Antidifferentiation

Evaluate the following integral

$\int (\sin(x)+\frac{x}{4})dx$

Possible Answers:

$-\cos(x)+\frac{x^2}{8}$

$-\cos(x)+\frac{x^2}{4}+C$

$-\cos(x)+\frac{x^2}{8}+C$

$\cos(x)+\frac{x^2}{8}+C$

Correct answer:

$-\cos(x)+\frac{x^2}{8}+C$

Explanation:

To solve the problem, we apply the fact that anti-derivative of $\sin(x)=-\cos(x)$ and that $\int x^n dx=\frac{x^{n+1}}{n+1}+C$

Taking the anti-derivative of each part independently, we get

$\int \sin(x)dx+\int \frac{x}{4}dx$

Finally, our answer is

$-\cos(x)+\frac{x^2}{8}+C$

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Example Question #651 : Ap Calculus Ab

Determine the value of $\int_{0}^{2}\sqrt{9x^2+12x+4} dx$ .

Possible Answers:

Correct answer:

Explanation:

We can factor the equation inside the square root:

$\\\int_0^2\sqrt{9x^2+12x+4}dx \\\\\int_0^2\sqrt{(3x+2)^2}dx \\\\\int_0^2 (3x+2)dx$

From here, increase each term's exponent by one and divide the term by the new exponent.

$\frac{3x^2}{2}+2x\big|_0^2$

Now, substitute in the upper bound into the function and subtract the lower bound function value from it.

$\\\frac{3(2)^2}{2}+2(2)-\left(\frac{3(0)^2}{2}+2(0)\right) \\\\\frac{3(4)}{2}+4 \\\\\frac{12}{2}+4 \\\\6+4 \\\\10$

Therefore,

$\int_{0}^{2}\sqrt{(3x+2)^2}dx=\int_{0}^{2}(3x+2)dx=10$

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Example Question #103 : Integrals

Evaluate the following integral

$\int (3x^3+2x-5 )dx$

Possible Answers:

$\frac{3}{4}x^3+x^4-5x^2+C$

$\frac{3}{4}x^4+x^2-5x+C$

Correct answer:

$\frac{3}{4}x^4+x^2-5x+C$

Explanation:

To evaluate the integral, we use the definition

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$

$\int (3x^3+2x-5 )dx=\frac{3}{4}x^4+x^2-5x+C$

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Example Question #104 : Integrals

Evaluate the following integral

$\int (\sin{x}+\sec^2{x)}dx$

Possible Answers:

$-\cos^2{x}+\tan^2{x}+C$

$\cos{x}+\tan{x}+C$

$-\cos{x}+\tan{x}$

$-\cos{x}+\tan{x}+C$

Correct answer:

$-\cos{x}+\tan{x}+C$

Explanation:

To evaluate the integral, we use the fact that the antiderivative of $\sin{x}$ is $-\cos{x}$ (because $\frac{\mathrm{d} }{\mathrm{d} x}(-\cos{x})=\sin{x}$ ), and the antiderivative of $\sec^2{x}$ is $\tan{x}$ (because $\frac{\mathrm{d} }{\mathrm{d} x}(\tan{x})=\sec^2x$ ). Using this information, we determine that the integral is

$-\cos{x}+\tan{x}+C$

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Example Question #105 : Integrals

Calculate the following integral.

$\int4sin(t)-12cos(t)+3t^2dt$

Possible Answers:

$-4cos(t)+12sin(t)+\frac{t^3}{3}+c$

4cos(t)+12sin(t)+t^4+c

-16tan(t)+t^3+c

-4cos(t)-12sin(t)+t^3+c

Correct answer:

-4cos(t)-12sin(t)+t^3+c

Explanation:

Calculate the following integral.

$\int4sin(t)-12cos(t)+3t^2dt$

To do this problem, we need to recall that integrals are also called antiderivatives. This means that we can calculate integrals by reversing our integration rules.

Thus, we can have the following rules.

$\int sin(t)dt = -cos(t)+c$

$\int cos(t)dt = sin(t)+c$

$\int t^ndt=\frac{t^{n+1}}{n+1}+c$

Using these rules, we can find our answer:

$\int4sin(t)-12cos(t)+3t^2dt$

Will become:

$-4cos(t)-12sin(t)+\frac{3t^3}{3}+c=-4cos(t)-12sin(t)+t^3+c$

And so our answer is:

-4cos(t)-12sin(t)+t^3+c

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AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #23 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #24 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #25 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #31 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #101 : Integrals

Example Question #34 : Techniques Of Antidifferentiation

Example Question #651 : Ap Calculus Ab

Example Question #103 : Integrals

Example Question #104 : Integrals

Example Question #105 : Integrals

All AP Calculus AB Resources

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