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Calculus 2 : Integrals

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Example Questions

Example Question #541 : Integrals

$\int_{1}^{2}3x^2+9dx$

Possible Answers:

Correct answer:

Explanation:

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

$\frac{3x^3}{3}+9x=x^3+9x$

Now, evaluate at 2 and then 1. Subtract the results:
(8+18)-(1+9)=26-10=16

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Example Question #542 : Integrals

$\int_{1}^{4}x^2+5x+4dx$

Possible Answers:

$\frac{101}{2}$

$\frac{149}{2}$

$\frac{142}{2}$

$\frac{141}{2}$

$\frac{141}{3}$

Correct answer:

$\frac{141}{2}$

Explanation:

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

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

Now, evaluate at 4 and then 1. Subtract the results:

$(\frac{64}{3}+40+16)-(\frac{1}{3}+\frac{5}{2}+4)$

Simplify to get your answer:

$21+56-4-\frac{5}{2}=\frac{141}{2}$

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Example Question #180 : Definite Integrals

$\int_{2}^{4}x^2-1dx$

Possible Answers:

$\frac{50}{3}$

$\frac{55}{3}$

$\frac{20}{3}$

$\frac{47}{3}$

$\frac{22}{3}$

Correct answer:

$\frac{50}{3}$

Explanation:

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:
$\frac{x^3}{3}-x$

Now evaluate at 4 and then 2. Subtract the results:
$(\frac{64}{3}-4)-(\frac{8}{3}-2)$

Simplify to get your answer:

$\frac{56}{2}-2=\frac{50}{3}$

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Example Question #181 : Definite Integrals

$\int_{0}^{1}6xe^{3x^2}dx$

Solve the definite integral by using u-substitution.

Possible Answers:

e^3-e

e^9-1

e^9-e

e^3-1

Correct answer:

e^3-1

Explanation:

So first things first, we identify what our u should be. If we look at this for chain rule our inside function would be the 3x^2 in the $e^{3x^2}$ . Therefore we use this as our u.

So we start with our u.

u=3x^2

Next we derive.

du=6xdx

Solve for dx.

$dx=\frac{du}{6x}$

Substitute it back in.

$\int_{0}^{1}6xe^u\frac{du}{6x}$

Simplify. If all the xs don't cross out, we have done something wrong.

$\int_{0}^{1}e^udu$

Integrate.

$e^u|_{0}^{1}$

Plug in the original.

$e^{3x^2}|_{0}^{1}$

Plug in values and substract

e^3-1

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Example Question #182 : Definite Integrals

Evaluate.

$\int_{1}^{2} 3 \ dx$

Possible Answers:

Answer not listed.

Correct answer:

Answer not listed.

Explanation:

$\int_{a}^{b}f(x) \ dx$

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

In this case, f(x) = 3 .

The antiderivative is F(x) = 3x .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{1}^{2} 3 \ dx = \left ( 3x \right )_{1}^{2}$

$= 3(2) - \left ( 3(1) \right )$

= 3

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Example Question #183 : Definite Integrals

Evaluate.

$\int_{0}^{1} 44x^{43} \ dx$

Possible Answers:

Answer not listed.

376,643,899

17,343,636

Correct answer:

Explanation:

$\int_{a}^{b}f(x) \ dx$

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

In this case, $f(x) = 44x^{43}$ .

The antiderivative is $F(x) = x^{44 }$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{0}^{1} 44x^{43} \ dx = \left ( x^{44 } \right )_{0}^{1}$

$= 1^{44 } - \left ( 0^{44 } \right )$

= 1

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Example Question #184 : Definite Integrals

Evaluate.

$\int_{7}^{8} \frac{1}{4x-19} \ dx$

Possible Answers:

Answer not listed.

Correct answer:

Answer not listed.

Explanation:

$\int_{a}^{b}f(x) \ dx$

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

In this case, $f(x) = \frac{1}{4x-19}$ .

The antiderivative is $F(x) = \frac{4x-19}{4}$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{7}^{8} \frac{1}{4x-19} \ dx = \left ( \frac{4x-19}{4} \right )_{7}^{8}$

$= \frac{4(8)-19}{4} - \left ( \frac{4(7)-19}{4} \right )$

= 1

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Example Question #185 : Definite Integrals

Evaluate.

$\int_{5}^{6} e^{8x-43}\ dx$

Possible Answers:

12.7

Answer not listed.

59.2

18.5

32.3

Correct answer:

18.5

Explanation:

$\int_{a}^{b}f(x) \ dx$

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

In this case, $f(x)= e^{8x-43}$ .

The antiderivative is $F(x) = \frac{e^{8x-43}}{8}$ .

Using the Fundamental Theorem of Calculus, evaluate the integral using the antiderivative:

$\int_{5}^{6} e^{8x-43}\ dx = \left ( \frac{e^{8x-43}}{8} \right )_{5}^{6}$

$= \frac{e^{8(6)-43}}{8} - \left ( \frac{e^{8(5)-43}}{8} \right )$

= 18.5

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Example Question #543 : Integrals

$\int_{0}^{2}x^2-\frac{1}{2}xdx$

Possible Answers:

$\frac{5}{3}$

$\frac{4}{3}$

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{10}{3}$

Correct answer:

$\frac{5}{3}$

Explanation:

First, integrate. Remember to raise the exponent by 1 and also put that result on the denominator:

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

Now, evaluate at 2 and then 0. Subtract the results:

$(\frac{2^3}{3})-\frac{2^2}{4}-0=$

Simplify to get your answer:

$\frac{8}{3}-\frac{3}{3}=\frac{5}{3}$

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Example Question #544 : Integrals

Evaluate the following:

$\int_{0}^{e}(\frac{1}{x}-2) dx$

Possible Answers:

ln(e-1)-e

2e+2

e-3

-2e-1

ln(2)-1

Correct answer:

-2e-1

Explanation:

Solving this problem first requires knowledge of antiderivatives and their rules as well as the properties of definite integrals.

$\int_{1}^{e}(\frac{1}{x}-2) dx= (ln(x)-2x)_{x=1}^{x=e}=$

=((ln(e)-2e)-(ln(1)-2))=(1-2e-2)=-2e-1

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Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #541 : Integrals

Example Question #542 : Integrals

Example Question #180 : Definite Integrals

Example Question #181 : Definite Integrals

Example Question #182 : Definite Integrals

Example Question #183 : Definite Integrals

Example Question #184 : Definite Integrals

Example Question #185 : Definite Integrals

Example Question #543 : Integrals

Example Question #544 : Integrals

All Calculus 2 Resources

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