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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #81 : Indefinite Integrals

Evaluate.

$\int x^6 - \frac{1}{3} x^{2} \ dx$

Possible Answers:

$F(x) =\frac{7x^6 - 3x^2}{3}$

$F(x) =\frac{x^7}{9} - \frac{x^3}{7}$

$F(x) =\frac{6x^7}{7} - \frac{3x^3}{9}$

Answer not listed

F(x) =7x^7 - 9x^3

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, $f(x) = x^6 - \frac{1}{3} x^{2}$ .

The antiderivative is $F(x) =\frac{x^7}{7} - \frac{x^3}{9}+C$ .

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Example Question #82 : Indefinite Integrals

Evaluate.

$\int \frac{3}{12x-5} \ dx$

Possible Answers:

$F(x) =\frac{3 \ln (12x-5)}{4}+C$

$F(x) =\frac{\ln (12x-5)}{12}+C$

$F(x) =\frac{\ln (12x-5)}{4}+C$

Answer not listed.

$F(x) =4 \ln (12x-5)$

Correct answer:

$F(x) =\frac{\ln (12x-5)}{4}+C$

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, $f(x) = \frac{3}{12x-5}$ .

The antiderivative is $F(x) =\frac{\ln (12x-5)}{4}+C$ .

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Example Question #83 : Indefinite Integrals

Evaluate.

$\int e^{9 - 2x}\ dx$

Possible Answers:

Answer not listed.

$F(x) =- e^{9 - 2x}+C$

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

$F(x) =-\frac{ e^{9 - 2x}}{2}+C$

$F(x) =\frac{ e^{9 - 2x}}{9}$

Correct answer:

$F(x) =-\frac{ e^{9 - 2x}}{2}+C$

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, $f(x) = e^{9 - 2x}$ .

The antiderivative is $F(x) =-\frac{ e^{9 - 2x}}{2}+C$ .

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Example Question #84 : Indefinite Integrals

Evaluate.

$\int \cos (2x-9) + 4 \ dx$

Possible Answers:

$F(x) =\frac{\sin (2x-9)}{2x} - 4x+C$

$F(x) =\frac{\sin (2x-9)}{2} - 4x+C$

$F(x) =2\sin (2x-9) - 4x+C$

Answer not listed.

$F(x) =\frac{\sin (2x-9)}{2} + 4x$

Correct answer:

$F(x) =\frac{\sin (2x-9)}{2} - 4x+C$

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, $f(x) = \cos (2x-9) + 4$ .

The antiderivative is $F(x) =\frac{\sin (2x-9)}{2} - 4x+C$ .

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Example Question #85 : Indefinite Integrals

Evaluate.

$\int \sec^2(5x-7) \ dx$

Possible Answers:

$F(x) = \frac{\sin (5x-7)}{5}$

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

Answer not listed.

$F(x) = \frac{\tan (5x-7)}{5}+C$

$F(x) = 5\tan (5x-7)+C$

Correct answer:

$F(x) = \frac{\tan (5x-7)}{5}+C$

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, $f(x) = \sec^2(5x-7)$ .

The antiderivative is $F(x) = \frac{\tan (5x-7)}{5}+C$ .

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Example Question #86 : Indefinite Integrals

$\int 4x^2-5x+1dx$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Remember, when integrating, raise the exponent of an x term by one and then put that result on the denominator.

Integrate each term separately.

Remember to add a C at the end because it is an indefinite integral.

Therefore, the answer is:

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

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Example Question #711 : Integrals

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

Possible Answers:

$\frac{5x^2}{2}+9x-ln\left | x \right |$

$\frac{x^2}{2}+9x-ln\left | x \right |+C$

$\frac{5x^2}{2}+9x-ln\left | x \right |+C$

$\frac{5x^2}{2}-9x-ln\left | x \right |+C$

$\frac{5x^2}{2}+9x+C$

Correct answer:

$\frac{5x^2}{2}+9x-ln\left | x \right |+C$

Explanation:

The first step here is to make the fraction three separate terms:

$\int 5x+9-\frac{1}{x}dx$ .

Then, integrate each term. Remember to raise the exponent of an x term by 1 and then put that result on the denominator.

Remember, if there is a single x on the denominator, integrating by taking ln of that term.

Therefore, the answer is:

$\frac{5x^2}{2}+9x-ln\left | x \right |+C$ .

Remember to add a +C at the end because it is an indefinite integral.

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Example Question #711 : Integrals

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Recall that when integrating, add one to the exponent and then put that result on the denominator:

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

Simplify and remember to add a +C because it is an indefinite integral.

Therefore, your answer is

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

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Example Question #91 : Indefinite Integrals

$\int 4x^3-2x-1dx$

Possible Answers:

x^4-x^2-x+C

x^4-2x^2-x+C

x^4-x^2-x

x^4-x^2+C

4x^4-x^2-x+C

Correct answer:

x^4-x^2-x+C

Explanation:

To integrate, remember to add one to the exponent and then put that result on the denominator.

The first step should look like this:

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

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

x^4-x^2-x+C .

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Example Question #713 : Integrals

Integrate:

$\int x\sin(2x)dx$

Possible Answers:

$\frac{-1}{2}x\cos(2x)+\frac{1}{4}\sin(2x)+C$

$\frac{-1}{2}x\cos(2x)-\frac{1}{4}\sin(2x)+C$

$\frac{-1}{2}\cos(2x)+\frac{1}{4}\sin(2x)+C$

$\frac{-1}{2}x\cos(2x)+\frac{1}{4}\sin(2x)$

Correct answer:

$\frac{-1}{2}x\cos(2x)+\frac{1}{4}\sin(2x)+C$

Explanation:

To integrate, we must integrate by parts, which states that

$\int udv=uv-\int vdu$

We must designate u and dv, and take their respective derivative and integral:

u=x , du=dx

$dv=\sin(2x)dx$ , $v=-\frac{1}{2}\cos(2x)$

The rules used are

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\int \sin(x)dx=-\cos(x)+C$

Next, we use the above formula to rewrite the integral:

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

We integrate again:

$\int \frac{1}{2}\cos(2x)dx=\frac{1}{4}\sin(2x)+C$

using the following rule:

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

Our final answer is

$\frac{-1}{2}x\cos(2x)+\frac{1}{4}\sin(2x)+C$

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #81 : Indefinite Integrals

Example Question #82 : Indefinite Integrals

Example Question #83 : Indefinite Integrals

Example Question #84 : Indefinite Integrals

Example Question #85 : Indefinite Integrals

Example Question #86 : Indefinite Integrals

Example Question #711 : Integrals

Example Question #711 : Integrals

Example Question #91 : Indefinite Integrals

Example Question #713 : Integrals

All Calculus 2 Resources

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