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Calculus 2 : Integrals

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Example Questions

Example Question #204 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{sin(x + 6)}{34}+\frac{ 1}{(11x)})dx\end{align*}$

Possible Answers:

$\frac{ln(x)}{1331}-\frac{ (18cos(x + 6))}{17}$

$\frac{ln(x)}{11}-\frac{ cos(x + 6)}{34}+C$

$\frac{ln(x)}{1331}-\frac{ (18cos(x + 6))}{17}+C$

$\frac{ln(x)}{11}-\frac{ cos(x + 6)}{34}$

Correct answer:

$\frac{ln(x)}{11}-\frac{ cos(x + 6)}{34}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\text{In deriving the inside of}\frac{sin(x + 6)}{34}\text{, the lone constant goes to zero.}\\&\text{The }a\text{ value for }\frac{1}{(11x)}\text{ is }\frac{1}{11}\\&\int(\frac{sin(x + 6)}{34}+\frac{ 1}{(11x)})dx=\frac{ln(x)}{11}-\frac{ cos(x + 6)}{34}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #205 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(3e^{(10x)})}{16}+\frac{ 1}{(116x)})dx\end{align*}$

Possible Answers:

$\frac{(15e^{(10x)})}{8}+\frac{ ln(x)}{1560896}$

$\frac{(3e^{(10x)})}{160}+\frac{ ln(x)}{116}+C$

$\frac{(15e^{(10x)})}{8}+\frac{ ln(x)}{1560896}+C$

$\frac{(3e^{(10x)})}{160}+\frac{ ln(x)}{116}$

Correct answer:

$\frac{(3e^{(10x)})}{160}+\frac{ ln(x)}{116}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\text{The }a\text{ value for }\frac{1}{(116x)}\text{ is }\frac{1}{116}\\&\text{The }a\text{ value for }\frac{(3e^{(10x)})}{16}\text{ is }10\\&\int(\frac{(3e^{(10x)})}{16}+\frac{ 1}{(116x)})dx=\frac{(3e^{(10x)})}{160}+\frac{ ln(x)}{116}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #831 : Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(58\cdot 2^{(\frac{x}{20})} +\frac{ x^{8}}{46})dx\end{align*}$

Possible Answers:

$\frac{(29\cdot 2^{(\frac{x}{20})})}{(10ln(2))}+\frac{ (32x^{9})}{207}+C$

$\frac{(29\cdot 2^{(\frac{x}{20})})}{(10ln(2))}+\frac{ (32x^{9})}{207}$

$\frac{(1160\cdot 2^{(\frac{x}{20})})}{ln(2)}+\frac{ x^{9}}{414}$

$\frac{(1160\cdot 2^{(\frac{x}{20})})}{ln(2)}+\frac{ x^{9}}{414}+C$

Correct answer:

$\frac{(1160\cdot 2^{(\frac{x}{20})})}{ln(2)}+\frac{ x^{9}}{414}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }58\cdot 2^{(\frac{x}{20})}\text{ is }\frac{1}{20}\\&\text{The }a\text{ value for }\frac{x^{8}}{46}\text{ is }8\\&\int(58\cdot 2^{(\frac{x}{20})} +\frac{ x^{8}}{46})dx=\frac{(1160\cdot 2^{(\frac{x}{20})})}{ln(2)}+\frac{ x^{9}}{414}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #207 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{cos(x + 14)}{47}+\frac{ (30e^{(2x)})}{7})dx\end{align*}$

Possible Answers:

$\frac{(60e^{(2x)})}{7}+\frac{ (196sin(x + 14))}{47}$

$\frac{(15e^{(2x)})}{7}+\frac{ sin(x + 14)}{47}$

$\frac{(15e^{(2x)})}{7}+\frac{ sin(x + 14)}{47}+C$

$\frac{(60e^{(2x)})}{7}+\frac{ (196sin(x + 14))}{47}+C$

Correct answer:

$\frac{(15e^{(2x)})}{7}+\frac{ sin(x + 14)}{47}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\text{The }a\text{ value for }\frac{(30e^{(2x)})}{7}\text{ is }2\\&\text{In deriving the inside of}\frac{cos(x + 14)}{47}\text{, the lone constant goes to zero.}\\&\int(\frac{cos(x + 14)}{47}+\frac{ (30e^{(2x)})}{7})dx=\frac{(15e^{(2x)})}{7}+\frac{ sin(x + 14)}{47}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #832 : Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{e^{(8x)}}{7}+\frac{ (7\cdot 2^{(\frac{x}{9})})}{2})dx\end{align*}$

Possible Answers:

$\frac{e^{(8x)}}{56}+\frac{ (63\cdot 2^{(\frac{x}{9})})}{(2ln(2))}$

$\frac{(8e^{(8x)})}{7}+\frac{ (7\cdot 2^{(\frac{x}{9})})}{(18ln(2))}$

$\frac{(8e^{(8x)})}{7}+\frac{ (7\cdot 2^{(\frac{x}{9})})}{(18ln(2))}+C$

$\frac{e^{(8x)}}{56}+\frac{ (63\cdot 2^{(\frac{x}{9})})}{(2ln(2))}+C$

Correct answer:

$\frac{e^{(8x)}}{56}+\frac{ (63\cdot 2^{(\frac{x}{9})})}{(2ln(2))}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\text{The }a\text{ value for }\frac{(7\cdot 2^{(\frac{x}{9})})}{2}\text{ is }\frac{1}{9}\\&\text{The }a\text{ value for }\frac{e^{(8x)}}{7}\text{ is }8\\&\int(\frac{e^{(8x)}}{7}+\frac{ (7\cdot 2^{(\frac{x}{9})})}{2})dx=\frac{e^{(8x)}}{56}+\frac{ (63\cdot 2^{(\frac{x}{9})})}{(2ln(2))}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #833 : Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(3sin(19x))}{5}+\frac{ (34\cdot 3^{(\frac{x}{11})})}{3})dx\end{align*}$

Possible Answers:

$\frac{(374\cdot 3^{(\frac{x}{11})})}{(3ln(3))}-\frac{ (3cos(19x))}{95}+C$

$\frac{(34\cdot 3^{(\frac{x}{11})})}{(33ln(3))}-\frac{ (57cos(19x))}{5}$

$\frac{(34\cdot 3^{(\frac{x}{11})})}{(33ln(3))}-\frac{ (57cos(19x))}{5}+C$

$\frac{(374\cdot 3^{(\frac{x}{11})})}{(3ln(3))}-\frac{ (3cos(19x))}{95}$

Correct answer:

$\frac{(374\cdot 3^{(\frac{x}{11})})}{(3ln(3))}-\frac{ (3cos(19x))}{95}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being added, we integrate separately}\\&\text{and combine the results. This is known as superposition.}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\text{The }a\text{ value for }\frac{(34\cdot 3^{(\frac{x}{11})})}{3}\text{ is }\frac{1}{11}\\&\text{The }a\text{ value for }\frac{(3sin(19x))}{5}\text{ is }19\\&\int(\frac{(3sin(19x))}{5}+\frac{ (34\cdot 3^{(\frac{x}{11})})}{3})dx=\frac{(374\cdot 3^{(\frac{x}{11})})}{(3ln(3))}-\frac{ (3cos(19x))}{95}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #210 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(28x^{12})}{5}-\frac{ x^{8}}{5})dx\end{align*}$

Possible Answers:

$\frac{(x^{9}\cdot (252x^{4} - 13))}{585}+C$

$\frac{(4032x^{13})}{65}-\frac{ (64x^{9})}{45}$

$\frac{(x^{9}\cdot (252x^{4} - 13))}{585}$

$\frac{(4032x^{13})}{65}-\frac{ (64x^{9})}{45}+C$

Correct answer:

$\frac{(x^{9}\cdot (252x^{4} - 13))}{585}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{(28x^{12})}{5}\text{ is }12\\&\text{The }a\text{ value for }\frac{x^{8}}{5}\text{ is }8\\&\int(\frac{(28x^{12})}{5}-\frac{ x^{8}}{5})dx=\frac{(x^{9}\cdot (252x^{4} - 13))}{585}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #834 : Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{67}{(7x)}- 27x^{13})dx\end{align*}$

Possible Answers:

$\frac{(300763ln(x))}{343}-\frac{ (4563x^{14})}{14}$

$\frac{(67ln(x))}{7}-\frac{ (27x^{14})}{14}$

$\frac{(300763ln(x))}{343}-\frac{ (4563x^{14})}{14}+C$

$\frac{(67ln(x))}{7}-\frac{ (27x^{14})}{14}+C$

Correct answer:

$\frac{(67ln(x))}{7}-\frac{ (27x^{14})}{14}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int\frac{a}{u}=aln(u)=ln(u^{a})\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{67}{(7x)}\text{ is }\frac{67}{7}\\&\text{The }a\text{ value for }27x^{13}\text{ is }13\\&\int(\frac{67}{(7x)}- 27x^{13})dx=\frac{(67ln(x))}{7}-\frac{ (27x^{14})}{14}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #212 : Indefinite Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(\frac{(23e^{(19x)})}{5}- 5x^{6})dx\end{align*}$

Possible Answers:

$\frac{(437e^{(19x)})}{5}-\frac{ (180x^{7})}{7}+C$

$\frac{(23e^{(19x)})}{95}-\frac{ (5x^{7})}{7}$

$\frac{(437e^{(19x)})}{5}-\frac{ (180x^{7})}{7}$

$\frac{(23e^{(19x)})}{95}-\frac{ (5x^{7})}{7}+C$

Correct answer:

$\frac{(23e^{(19x)})}{95}-\frac{ (5x^{7})}{7}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\text{The }a\text{ value for }\frac{(23e^{(19x)})}{5}\text{ is }19\\&\text{The }a\text{ value for }5x^{6}\text{ is }6\\&\int(\frac{(23e^{(19x)})}{5}- 5x^{6})dx=\frac{(23e^{(19x)})}{95}-\frac{ (5x^{7})}{7}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Example Question #491 : Finding Integrals

$\begin{align*}&\text{Evaluate the indefinite integral: }\\&\int(23e^{(15x)} -\frac{ (17\cdot 3^{(20x)})}{4})dx\end{align*}$

Possible Answers:

$\frac{(23e^{(15x)})}{15}-\frac{ (17\cdot 3^{(20x)})}{(80ln(3))}+C$

$\frac{(23e^{(15x)})}{15}-\frac{ (17\cdot 3^{(20x)})}{(80ln(3))}$

$345e^{(15x)} -\frac{ (85\cdot 3^{(20x)})}{ln(3)}$

$345e^{(15x)} -\frac{ (85\cdot 3^{(20x)})}{ln(3)}+C$

Correct answer:

$\frac{(23e^{(15x)})}{15}-\frac{ (17\cdot 3^{(20x)})}{(80ln(3))}+C$

Explanation:

$\begin{align*}&\text{In performing an integral, consider how in a derivative we’d}\\&\text{use the chain rule to multiply the derivative of the “inside”}\\&\text{of a function by the derivative of the outside. Integrals, in}\\&\text{similar fashion, entail a division. Now, although we have two}\\&\text{functions, since they're being subtracted (rather than multiplied/divided),}\\&\text{we integrate separately and combine the results. This is known}\\&\text{as superposition.}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\text{The }a\text{ value for }23e^{(15x)}\text{ is }15\\&\text{The }a\text{ value for }\frac{(17\cdot 3^{(20x)})}{4}\text{ is }20\\&\int(23e^{(15x)} -\frac{ (17\cdot 3^{(20x)})}{4})dx=\frac{(23e^{(15x)})}{15}-\frac{ (17\cdot 3^{(20x)})}{(80ln(3))}+C\\&\text{Now you'll note after integration we add a constant, C, because}\\&\text{we're working with an indefinite integral.}\\&\end{align*}$

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #204 : Indefinite Integrals

Example Question #205 : Indefinite Integrals

Example Question #831 : Integrals

Example Question #207 : Indefinite Integrals

Example Question #832 : Integrals

Example Question #833 : Integrals

Example Question #210 : Indefinite Integrals

Example Question #834 : Integrals

Example Question #212 : Indefinite Integrals

Example Question #491 : Finding Integrals

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