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Calculus 2 : Definite Integrals

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Example Questions

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^3-1

e^9-e

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:

17,343,636

Answer not listed.

376,643,899

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:

Answer not listed.

32.3

12.7

59.2

18.5

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 #186 : Definite Integrals

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

Possible Answers:

$\frac{10}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{1}{3}$

$\frac{5}{3}$

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 #187 : Definite Integrals

Evaluate the following:

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

Possible Answers:

2e+2

-2e-1

ln(e-1)-e

e-3

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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Example Question #188 : Definite Integrals

Solve the following definite integral.

$\int_0^5 \,2xy+3\,dx$

Possible Answers:

25y+15

x^2y+3x

xy^2+3y

Correct answer:

25y+15

Explanation:

$\int_0^5 \,2xy+3\,dx=x^2y+3x|_{x=0}^{x=5}$

=5^2y+3(5)-[0^2y+3(0)]

=25y+15

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Example Question #2302 : Calculus Ii

If $\int_{1}^{7}f(x)dx=5, \int_{3}^{7}f(x)dx=2, \int_{1}^{3}f(x)dx=?$

Possible Answers:

Not enough information.

Correct answer:

Explanation:

If we think of the anti-derivative as computing the area under the curve, then between x=1 and x=7 , the area under our function equals . We also know that between x=3 and x=7 , the area under our function equals . Then, to find the area under the function between x=1 and x=3 , we must simply subtract the other two areas. Therefore:

$\int_{1}^{7}f(x)dx- \int_{3}^{7}f(x)dx=\int_{1}^{3}f(x)dx=5-2=3.$

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Example Question #189 : Definite Integrals

If f(x)=x^2-3x+5 , what is $\int_{5}^{5}f(x)dx$ ?

Possible Answers:

$\frac{1}{2}$

$\frac{2}{3}$

Correct answer:

Explanation:

Remember that the anti-derivative computes the area under the curve of the function between the values specified by the upper and lower integral limits. In this question, the upper and lower integral limits match! This, by definition, means that the integral equals There is no area if you start and end at the same point!

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Calculus 2 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

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 #186 : Definite Integrals

Example Question #187 : Definite Integrals

Example Question #188 : Definite Integrals

Example Question #2302 : Calculus Ii

Example Question #189 : Definite Integrals

All Calculus 2 Resources

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