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Calculus 2 : Integrals

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9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #121 : Indefinite Integrals

Integrate:

$\int (4x+\frac{1}{16+x^2})dx$

Possible Answers:

$2x^2+\arctan({2x})+C$

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

$2x^2+\frac{\arctan(\frac{x}{4})}{4}$

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

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

Correct answer:

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

Explanation:

The integral is equal to

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

and was found using the following rules:

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$ , $\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan(\frac{x}{a})+C$

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Example Question #122 : Indefinite Integrals

$\int x\sqrt{x^2-1}dx$

Possible Answers:

$\frac{(x^2-1)^{\frac{3}{2}}}{3}+C$

$\frac{(x^2-1)^{\frac{3}{2}}}{3}$

$\frac{(x^2-1)^{\frac{2}{3}}}{3}+C$

$\frac{(x-1)^{\frac{3}{2}}}{3}+C$

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

Correct answer:

$\frac{(x^2-1)^{\frac{3}{2}}}{3}+C$

Explanation:

To integrate this expression, use u substitution:

$u=x^2-1; du=2xdx; \frac{1}{2}du=xdx$

Now, substitute in everything and rewrite the expression:

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

Integrate and remember to raise the exponent by 1 and then put that result on the denominator:

$(\frac{1}{2})(\frac{u^{\frac{3}{2}}}{\frac{3}{2}})$

Simplify and sub back in your initial expression:

$\frac{u^{\frac{3}{2}}}{3}=\frac{(x^2-1)^{\frac{3}{2}}}{3}$

Remember to add C because it is an indefinite integral:
$\frac{(x^2-1)^{\frac{3}{2}}}{3}+C$ .

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Example Question #123 : Indefinite Integrals

$\int 2x\sqrt{x^2-4}dx$

Possible Answers:

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

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

$\frac{2}{3}(x^2-4)^{\frac{1}{2}}+C$

$\frac{1}{3}(x^2-4)^{\frac{3}{2}}+C$

$\frac{2}{3}(x^2-4)^{\frac{3}{2}}$

Correct answer:

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

Explanation:

To integrate this expression, use u substitution:

u=x^2-4; du=2xdx

Now, substitute everything in and rewrite the expression:

$\frac{u^{\frac{3}{2}}}{\frac{3}{2}}$

Now, simplify and sub back in your initial expression:

$\frac{2}{3}(x^2-4)^{\frac{3}{2}}$

Don't forget to add a C because it is an indefinite integral:

$\frac{2}{3}(x^2-4)^{\frac{3}{2}}+C$ .

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Example Question #124 : Indefinite Integrals

$\int \frac{4x^2-16x+12}{4}dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

First, chop up this fraction into three separate terms so you can more easily integrate:

$\int x^2-4x+3dx$

Now, integrate. Remember to add one to the exponent and also put that result on the denominator:

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

Simplify and add a C because it is an indefinite integral:

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

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Example Question #125 : Indefinite Integrals

$\int 2x^2+x-1dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

To integrate, remember to raise the exponent by 1 and then put that result on the denominator:

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

Remember to add a C because it is an indefinite integral:

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

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Example Question #756 : Integrals

Evaluate.

$\int 13i \ dx$

Possible Answers:

F(x) = 13ix^2 +C

F(x) = 13ix +C

F(x) = 13i +C

F(x) = ix +C

Answer not listed

Correct answer:

F(x) = 13ix +C

Explanation:

$\int f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, f(x) = 13i .

The antiderivative is F(x) = 13ix +C .

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Example Question #757 : Integrals

Evaluate.

$\int \frac{x}{3-x} \ dx$

Possible Answers:

$F(x) = - \ln(x) - x+C$

$F(x) = -3 \ln(3-x)+C$

Answer not listed

$F(x) = \ln(3-x)+C$

$F(x) = \ln(3-x) + 3x+C$

Correct answer:

Answer not listed

Explanation:

$\int f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \frac{x}{3-x}$ .

The antiderivative is $F(x) = -3 \ln(3-x) - x+C$ .

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Example Question #751 : Integrals

Evaluate.

$\int e^{2x-5} \ dx$

Possible Answers:

$F(x) = e^{2x-5} +C$

$F(x) = \frac{e^{2x-5} }{2} +C$

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

$F(x) = 2e^{2x-5} +C$

Answer not listed.

Correct answer:

$F(x) = \frac{e^{2x-5} }{2} +C$

Explanation:

$\int f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = e^{2x-5}$ .

The antiderivative is $F(x) = \frac{e^{2x-5} }{2} +C$ .

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Example Question #752 : Integrals

Evaluate.

$\int \sin (5x-3) \ dx$

Possible Answers:

Answer not listed.

$F(x) = -\frac{ \cos(5x-3) }{5x-3} +C$

$F(x) = \frac{ \cos(5x-3) }{x} +C$

$F(x) = -\frac{ \sin(5x) }{5} +C$

$F(x) = -\frac{ \cos(5x-3) }{5} +C$

Correct answer:

$F(x) = -\frac{ \cos(5x-3) }{5} +C$

Explanation:

$\int f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \sin (5x-3)$ .

The antiderivative is $F(x) = -\frac{ \cos(5x-3) }{5} +C$ .

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Example Question #133 : Indefinite Integrals

Evaluate.

$\int \sec^2(3x) \ dx$

Possible Answers:

$F(x) = \frac{ \tan(3x) }{x} +C$

Answer not listed.

$F(x) = \frac{ \tan^3(x) }{3} +C$

$F(x) = \frac{ \cos(3) }{x} +C$

$F(x) = \frac{ \tan(x) }{3} +C$

Correct answer:

Answer not listed.

Explanation:

$\int f(x) dx$

In order to evaluate this integral, first find the antiderivative of f(x).

If f(x) = a then F(x) = ax + C

If f(x) = x a then $F(x) = \frac{x^{a+1}}{a+1} + C$

If $f(x) = \frac{1}{x}$ then F(x) = ln(x) + C

If f(x) = e ^x then F(x) = e^x + C

If $f(x) = \cos(x)$ then $F(x) = \sin(x) + C$

If $f(x) = \sin(x)$ then $F(x) = - \cos(x) + C$

If $f(x) = \sec^2 (x)$ then $F(x) = \tan(x) + C$

In this case, $f(x) = \sec^2(3x)$ .

The antiderivative is $F(x) = \frac{ \tan(3x) }{3} +C$ .

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #121 : Indefinite Integrals

Example Question #122 : Indefinite Integrals

Example Question #123 : Indefinite Integrals

Example Question #124 : Indefinite Integrals

Example Question #125 : Indefinite Integrals

Example Question #756 : Integrals

Example Question #757 : Integrals

Example Question #751 : Integrals

Example Question #752 : Integrals

Example Question #133 : Indefinite Integrals

All Calculus 2 Resources

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