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AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

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Example Questions

Example Question #11 : Antiderivatives Following Directly From Derivatives Of Basic Functions

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-6}^{-2.5}(10e^{(x)})dx\end{align*}$

Possible Answers:

0.14

0.23

0.80

6.61

Correct answer:

0.80

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{-6}^{-2.5}(10e^{(x)})dx=10e^{(x)}|_{-6}^{-2.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-6}^{-2.5}(10e^{(x)})dx=10e^{((-2.5))}-(10e^{((-6))})\\&\int_{-6}^{-2.5}(10e^{(x)})dx=0.80\end{align*}$

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Example Question #632 : Ap Calculus Ab

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{7.5}^{12.5}(\frac{3^{(\frac{x}{3})}}{4})dx\end{align*}$

Possible Answers:

55.77

540.93

223.07

340.17

Correct answer:

55.77

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{7.5}^{12.5}(\frac{3^{(\frac{x}{3})}}{4})dx=\frac{(3\cdot 3^{(\frac{x}{3})})}{(4ln(3))}|_{7.5}^{12.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{7.5}^{12.5}(\frac{3^{(\frac{x}{3})}}{4})dx=\frac{(3\cdot 3^{(\frac{(12.5)}{3})})}{(4ln(3))}-(\frac{(3\cdot 3^{(\frac{(7.5)}{3})})}{(4ln(3))})\\&\int_{7.5}^{12.5}(\frac{3^{(\frac{x}{3})}}{4})dx=55.77\end{align*}$

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Example Question #633 : Ap Calculus Ab

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{14}^{15}(\frac{2^{(\frac{x}{3})}}{20})dx\end{align*}$

Possible Answers:

1.43

0.16

0.24

3.00

Correct answer:

1.43

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{14}^{15}(\frac{2^{(\frac{x}{3})}}{20})dx=\frac{(3\cdot 2^{(\frac{x}{3})})}{(20ln(2))}|_{14}^{15}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{14}^{15}(\frac{2^{(\frac{x}{3})}}{20})dx=\frac{(3\cdot 2^{(\frac{(15)}{3})})}{(20ln(2))}-(\frac{(3\cdot 2^{(\frac{(14)}{3})})}{(20ln(2))})\\&\int_{14}^{15}(\frac{2^{(\frac{x}{3})}}{20})dx=1.43\end{align*}$

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Example Question #634 : Ap Calculus Ab

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{9}^{12.5}(\frac{(20cos(3x))}{3})dx\end{align*}$

Possible Answers:

-2.56

-20.52

-5.64

-0.42

Correct answer:

-2.56

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int_{9}^{12.5}(\frac{(20cos(3x))}{3})dx=\frac{(20sin(3x))}{9}|_{9}^{12.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{9}^{12.5}(\frac{(20cos(3x))}{3})dx=\frac{(20sin(3(12.5)))}{9}-(\frac{(20sin(3(9)))}{9})\\&\int_{9}^{12.5}(\frac{(20cos(3x))}{3})dx=-2.56\end{align*}$

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Example Question #635 : Ap Calculus Ab

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{10}^{15}(\frac{51}{(2x)})dx\end{align*}$

Possible Answers:

55.83

10.34

1.26

36.19

Correct answer:

10.34

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int_{10}^{15}(\frac{51}{(2x)})dx=\frac{(51ln(x))}{2}|_{10}^{15}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{10}^{15}(\frac{51}{(2x)})dx=\frac{(51ln((15)))}{2}-(\frac{(51ln((10)))}{2})\\&\int_{10}^{15}(\frac{51}{(2x)})dx=10.34\end{align*}$

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Example Question #21 : Techniques Of Antidifferentiation

$\begin{align*}&\text{Evaluate the integral: }\\&\int_{-12.5}^{-11}(\frac{23}{(4\cdot 3^{(\frac{x}{2})})})dx\end{align*}$

Possible Answers:

34950.60

5637.19

626.35

1761.62

Correct answer:

5637.19

Explanation:

$\begin{align*}&\text{In performing an integral, familiarity with derivative rules}\\&\text{can prove useful. After all, an integral is essentially the}\\&\text{opposite operation of a derivative. Consider how in a derivative}\\&\text{we’d use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside? Integrals, in}\\&\text{similar fashion, entail a division.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{-12.5}^{-11}(\frac{23}{(4\cdot 3^{(\frac{x}{2})})})dx=-\frac{23}{(2\cdot 3^{(\frac{x}{2})}ln(3))}|_{-12.5}^{-11}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{-12.5}^{-11}(\frac{23}{(4\cdot 3^{(\frac{x}{2})})})dx=-\frac{23}{(2\cdot 3^{(\frac{(-11)}{2})}ln(3))}-(-\frac{23}{(2\cdot 3^{(\frac{(-12.5)}{2})}ln(3))})\\&\int_{-12.5}^{-11}(\frac{23}{(4\cdot 3^{(\frac{x}{2})})})dx=5637.19\end{align*}$

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Example Question #22 : Techniques Of Antidifferentiation

Given f'(x) = (2x+5)^2 , find the general form for the antiderivative f(x) .

Possible Answers:

8x+20+C

None of the other answers

4x^2+20x+25+C

$\frac{4}{3}x^3 +10x^2 +25x+C$

8x+10+C

Correct answer:

$\frac{4}{3}x^3 +10x^2 +25x+C$

Explanation:

To answer this, we will need to FOIL our function first.

(2x+5)^2 = 4x^2+20x+25

Now can find the antiderivatives of each of these three summands using the power rule.

$f(x) = \frac{4}{3}x^3 +10x^2 +25x+C$ (Don't forget )!

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Example Question #21 : Techniques Of Antidifferentiation

Compute the following integral:

$\int 6t^5-12t+6e^t-sin(t)+14cos(t)dt$

Possible Answers:

t^6-6t^2+6e^t+cos(t)+14sin(t)+c

t^6-6t^2+6e^t+cos(t)+14sin(t)+5c

t^6-6t^2+6e^t+cos(t)+14sin(t)

t^6-6t^2+6e^t-cos(t)-14sin(t)+c

Correct answer:

t^6-6t^2+6e^t+cos(t)+14sin(t)+c

Explanation:

Compute the following integral:

$\int 6t^5-12t+6e^t-sin(t)+14cos(t)dt$

Now, we need to recall a few rules.

$\int e^xdx=e^x+c$

$\int x^ndx=\frac{x^{n+1}}{n+1}+c$

$\int sin(x)dx=-cos(x)+c$

$\int cos(x)dx=sin(x)+c$

We can use all these rules to change our original function into its anti-derivative.

$\int 6t^5-12t+6e^t-sin(t)+14cos(t)dt$

We can break this up into separate integrals for each term, and apply our rules individually.

$\int 6t^5dt-\int 12tdt+\int 6e^tdt-\int sin(t)dt+\int 14cos(t)dt$

The first two integrals can be found using rule 2

$\int 6t^5dt-\int 12tdt=\frac{6t^6}{6}+c-\frac{12t^2}{2}+c=t^6-6t^2+c$

Next, let's tackle the middle integral:

$\int 6e^tdt=6e^t +c$

Then the "sine" integral

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

And finally, the cosine integral.

$\int 14cos(t)dt=14sin(t)+c$

Now, we can put all of this together to get:

t^6-6t^2+6e^t+cos(t)+14sin(t)+c

Note that we only have 1 c, because the c is just a constant.

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Example Question #21 : Techniques Of Antidifferentiation

Solve:

$\int (5\sec^2(x)+\sec(x)\tan(x))dx$

Possible Answers:

$5\tan(x)+\ln(\sec(x))+C$

$5(\tan(x)+\sec(x))+C$

$5\tan(x)+\sec(x)+C$

$5\tan(x)+\sec(x)$

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

Correct answer:

$5\tan(x)+\sec(x)+C$

Explanation:

The integral can be solved knowing the derivatives of the following functions:

$\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$

Given that the integrand is simply the sum of these two derivatives, we find that our integral is equal to

$\int (5\sec^2(x)+\sec(x)\tan(x))dx=5\tan(x)+\sec(x)+C$

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Example Question #641 : Ap Calculus Ab

Solve:

$\int \frac{5}{x^3}dx$

Possible Answers:

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

$-\frac{5}{2x^2}$

$5\ln\left | x^2\right |+C$

None of the other answers

Correct answer:

None of the other answers

Explanation:

The integral is equal to

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

and was given by the following rule:

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

Using this rule becomes more clear when we rewrite the integral as

$\int 5x^{-3}dx$

Note that because none of the answer choices had the integration constant C along with the proper integral result, the correct choice was "None of the other answers." Always check after solving an indefinite integral for C!

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AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #11 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Example Question #632 : Ap Calculus Ab

Example Question #633 : Ap Calculus Ab

Example Question #634 : Ap Calculus Ab

Example Question #635 : Ap Calculus Ab

Example Question #21 : Techniques Of Antidifferentiation

Example Question #22 : Techniques Of Antidifferentiation

Example Question #21 : Techniques Of Antidifferentiation

Example Question #21 : Techniques Of Antidifferentiation

Example Question #641 : Ap Calculus Ab

All AP Calculus AB Resources

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