Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #961 : Integrals

Evaluate the following indefinite integral using the substitution method.


Possible Answers:

Correct answer:

Explanation:

The integral can be expanded by distributing the exponent.

 

We will make the following substitution:

.

 

Differentiating both sides yields

.

 

We can then substitute the left hand side of each equation into our integral and evaluate it now.

 

Finally, we substitute the original variable back into the expression:

.

 

Example Question #961 : Integrals

Solve:

 

Possible Answers:

Correct answer:

Explanation:

Use substitution:

Plug the  and  into the regular equation, but no need to worry about the bounds yet:

 

Plug  back into the integrated equation from above and evaluate from  to .

 

Example Question #2711 : Calculus Ii

Solve:

 

Possible Answers:

None of the chocies.

Correct answer:

Explanation:

Use substitution integration:

 

Example Question #25 : Solving Integrals By Substitution

What is the integral of ?

Possible Answers:

Correct answer:

Explanation:

Use substitution:

 

    

Substitute  back in.

 

Example Question #962 : Integrals

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we must first perform the following substitution:

Now, rewrite the integral and integrate:

The integration was performed using the following rule:

Finally, replace u with the original term:

Example Question #963 : Integrals

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we must make the following substitution:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with our original term:

Example Question #31 : Solving Integrals By Substitution

Evaluate the indefinite integral .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We proceed as follows,

. Start

. Factor out the 10.

 

Use u-substitution with , then taking derivates of both sides gives.

 

. Substitute values

. Factor out the negative.

. The antiderivative of  is . Don't forget .

. Substitute  back.

 

Example Question #32 : Solving Integrals By Substitution

Evaluate the indefinite integral .

Possible Answers:

None of the other answers

Not possible to integrate

Correct answer:

Explanation:

We evaluate the integral as follows,

 

. Start

Use u-substitution, let , then taking derivatives of both sides gives . Divide both sides of this equation by , giving . Now we can substitute out , and get

 

. Factor out the .

. Integrate  and add .

. Substitute back

Example Question #35 : Solving Integrals By Substitution

Evaluate .

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We use u-substitution to evaluate this integral.

Let . Subtracting  gives , and taking derivatives gives (We subtract  from both sides in order to make the expression under the square root as simple as possible). Then we have

 

. Start

. Make our substitutions. (Make sure you change the bounds of integration too, by plugging  and  into  for ).

.

Example Question #31 : Solving Integrals By Substitution

Evaluate 

Possible Answers:

None of the other answers.

Correct answer:

None of the other answers.

Explanation:

The correct answer is .

 

We proceed as follows-

 

. Start

Evaluating this integral relies on the fact , and the Chain Rule for derivatives.

 

Use u-substitution , then we obtain 

Our integral then becomes

 after substitution. (The new upper bound on the integral cannot be simplified well, so we should leave it as is).

We then integrate to get

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