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Calculus 2 : Integrals

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Example Questions

Example Question #761 : Integrals

Evaluate the integral:

$\int \frac{x^4+2x^2+5}{x}+\frac{1}{x^2+9}dx$

Possible Answers:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\arctan(\frac{x}{3})+C$

$\frac{x^3}{3}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arcsin(\frac{x}{3})+C$

Correct answer:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

Explanation:

To evaluate the integral, it is easiest if we split it into two integrals:

$\int \frac{x^4+2x^2+5}{x}dx+\int \frac{dx}{x^2+9}$

To integrate the first integral, we can divide to rewrite the integrand into something easier to integrate:

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

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

The following rules were used to integrate:

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

Next, we integrate the second integral

$\int \frac{dx}{x^2+9}=\frac{1}{3}\arctan(\frac{x}{3})+C$

using the following rule:

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

Finally, combine the two results together:

$\frac{x^4}{4}+x^2+5\ln\left | x\right |+\frac{1}{3}\arctan(\frac{x}{3})+C$

Note that the constants of integration combine to make a single constant.

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Example Question #762 : Integrals

Evaluate the following integral: $\int(2n^2+5n+1)dn$

Possible Answers:

4n+5+C

$\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

$2n-\frac{5}{n}-n+C$

2n^3+5n^2+n+C

Correct answer:

$\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

Explanation:

To evaluate the integral, we use the following rule: $\int(x^n)dx=\frac{x^{n+1}}{n+1}+C$ . Applying the rule, we get $\int(2n^2+5n+1)dn$ = $\frac{2n^3}{3}+\frac{5n^2}{2}+n+C$

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Example Question #763 : Integrals

Evaluate the following integral: $\int (\sin(x)+3x^2)dx$

Possible Answers:

$-\cos(x)+6x+C$

$\sin(x)+x^3+C$

$\cos(x)+x^3+C$

$-\cos(x)+x^3+C$

Correct answer:

$-\cos(x)+x^3+C$

Explanation:

Using the rules for integration, $\int x^ndx=\frac{x^{n+1}}{n+1}+C$ and $\int \sin(x)= -\cos(x)+C$ , the integral is evaluated. For the first part, we established the value, and for the term involving the exponent, $\int 3x^2dx=3x^3/3=x^3+C$ . We put these two things together to get the correct answer.

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Example Question #136 : Indefinite Integrals

Evaluate.

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

Possible Answers:

Answer not listed.

F(x) = 4 x^2 + C

$F(x) = \frac{1}{4} x^2 + C$

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

F(x) = 2 x^2 + C

Correct answer:

$F(x) = \frac{1}{4} x^2 + C$

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, $f(x) = \frac{1}{2} x$ .

The antiderivative is $F(x) = \frac{1}{4} x^2 + C$ .

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Example Question #137 : Indefinite Integrals

Evaluate.

$\int e^ {x -7}\ dx$

Possible Answers:

$F(x) =\frac{e^x}{7} +C$

$F(x) = 7e^ {x} + C$

$F(x) = e^ {x -7} + C$

$F(x) = e^ {7x} + C$

Answer not listed.

Correct answer:

$F(x) = e^ {x -7} + C$

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, $f(x) = e^ {x -7}$ .

The antiderivative is $F(x) = e^ {x -7} + C$ .

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Example Question #138 : Indefinite Integrals

Evaluate.

$\int \frac{x-4}{x-5} \ dx$

Possible Answers:

$F(x) =\ln(x-5) + x + C$

$F(x) =5\ln(x-4) + x + C$

Answer not listed.

$F(x) =5\ln(x) + x + C$

$F(x) =\ln(x-5)(x-4) + C$

Correct answer:

$F(x) =\ln(x-5) + x + C$

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, $f(x) =\frac{x-4}{x-5}$ .

The antiderivative is $F(x) =\ln(x-5) + x + C$ .

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Example Question #764 : Integrals

$\int \frac{2t-1}{t}dt$

Possible Answers:

$t-ln\left | t \right |+C$

$2t-2ln\left | t \right |+C$

$2t-ln\left | t \right |+C$

$2t-ln\left | t \right |$

$2t^2-ln\left | t \right |+C$

Correct answer:

$2t-ln\left | t \right |+C$

Explanation:

First, chop up the fraction into two separate terms:

$\int 2-\frac{1}{t}dt$

Then, integrate.

$2t-ln\left | t \right |$

Now, remember to add a C because it is an indefinite integral:

$2t-ln\left | t \right |+C$ .

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Example Question #419 : Finding Integrals

$\int 5t^2-2t+6dt$

Possible Answers:

$\frac{5}{3}t^3-t^2-6t+C$

$\frac{5}{3}t^3-t^2+6t+C$

$\frac{5}{3}t^3-t^2+6t$

$\frac{1}{3}t^3-t^2+6t+C$

$\frac{5}{3}t^3+t^2+6t+C$

Correct answer:

$\frac{5}{3}t^3-t^2+6t+C$

Explanation:

Integrate this expression. Remember to add one to the exponent and also put that result on the denominator:

$5(\frac{t^3}{3})-2(\frac{t^2}{2})+6t$

Simplify and add C because it is an indefinite integral:

$\frac{5}{3}t^3-t^2+6t+C$ .

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Example Question #420 : Finding Integrals

$\int x^{\frac{4}{3}}dx$

Possible Answers:

$\frac{3}{7}x^{\frac{1}{3}}+C$

$\frac{3}{7}x^{\frac{7}{3}}+C$

$\frac{4}{7}x^{\frac{7}{3}}+C$

$\frac{2}{7}x^{\frac{7}{3}}+C$

$\frac{3}{8}x^{\frac{7}{3}}+C$

Correct answer:

$\frac{3}{7}x^{\frac{7}{3}}+C$

Explanation:

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

$\frac{x^{\frac{7}{3}}}{\frac{7}{3}}$

Simplify and add C because it is an indefinite integral:

$\frac{3}{7}x^{\frac{7}{3}}+C$

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Example Question #421 : Finding Integrals

$\int \frac{4x^3}{x^4+3}dx$

Possible Answers:

$ln\left | 4x^3 \right |+C$

$ln\left | x^4-3 \right |+C$

$ln\left | x^4+3 \right |$

$2ln\left | x^4+3 \right |+C$

$ln\left | x^4+3 \right |+C$

Correct answer:

$ln\left | x^4+3 \right |+C$

Explanation:

First, you must use u substitution to integrate this expression:

u=x^4+3; du=4x^3dx

Now, plug back in so you can integrate:

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

Now integrate:

$ln\left | u \right |$

Now substitute back in your initial expression and add C because it is an indefinite integral:

$ln\left | x^4+3 \right |+C$

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #761 : Integrals

Example Question #762 : Integrals

Example Question #763 : Integrals

Example Question #136 : Indefinite Integrals

Example Question #137 : Indefinite Integrals

Example Question #138 : Indefinite Integrals

Example Question #764 : Integrals

Example Question #419 : Finding Integrals

Example Question #420 : Finding Integrals

Example Question #421 : Finding Integrals

All Calculus 2 Resources

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