Calculus 2 : Definite Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #21 : Definite Integrals

Evaluate the indefinite integral .

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is .

 

The integral itself is not too difficult to take, simply use the Power Rule on the  and  terms. The trick is to be careful when integrating  is a constant value (about ) not a variable, so it must be integrated accordingly.

Example Question #21 : Definite Integrals

Evaluate  .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

This integral requires integration by parts followed by u-substitution. Here are the details

. Start

. Factor out the 

Set up integration by parts with . We then have  and . Afterward, we use the integration by parts formula .

.

Now at this point we use u-substitution to evaluate the 2nd integral. Let , then  and therefore . Substituting into the integral we have

. (Don't forget to change the bounds of integration by plugging them into  for our equation for .)

 

Example Question #381 : Integrals

Evaluate .

Possible Answers:

Not possible without a calculator

Correct answer:

Explanation:

This integral isn't possible to integrate directly using antiderivatives, but we can still find its value by noticing that is an odd function , and that our limits of integration are negatives of each other.

Hence

. (Since is an odd function)

Example Question #31 : Finding Integrals

Evaluate

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We can use u-substitution for this integral

. Start

Let , then , and our integral becomes

. (Don't forget to change the bounds of integration by plugging them into our equation for )

 

Example Question #22 : Definite Integrals

Evaluate .

Possible Answers:

None of the other answers

Correct answer:

Explanation:

We proceed by using integration by parts.

. Start

Let , then we get 
. Then using the integration by parts formula , we get

 

 

 

Example Question #26 : Definite Integrals

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate this integral, we must integrate by parts, according to the following formula:

So, we must assign our u and dv, and differentiate and integrate to find du and v, respectively:

The derivative and integral were found using the following rules:

Note that we ignore the constant of integration.

Now, use the above formula:

Note that both the product of u and v and the integral are being evaluated from zero to 

The integral was performed using the following rule:

Simplifying the above results, we get .

Example Question #21 : Definite Integrals

Find the area between  and  between 

Possible Answers:

Correct answer:

Explanation:

We can write this problem as:

Integrating: 

By the fundamental theorem of calculus:

Example Question #391 : Integrals

Possible Answers:

Correct answer:

Explanation:

Compute the Indefinite Integral

Evaluate the integral

Example Question #29 : Definite Integrals

Suppose , where  is a constant 

Find  such that 

Possible Answers:

Correct answer:

Explanation:

By the fundamental theorem of calculus:

 

Example Question #21 : Definite Integrals

Evaluate this integral.

Possible Answers:

Answer not listed

Correct answer:

Explanation:

In order to evaluate this integral, first find the antiderivative of 

In this case, .

The antiderivative is  .

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

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