Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #72 : Solving Integrals By Substitution

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

A u-substitution would properly simplify the integrand, where 

Now, the problem can be rewritten entirely in terms of u:

The problem may seem finished, but the original integrand was expressed in terms of . Therefore, the final answer is, in fact:

 

Example Question #77 : Solving Integrals By Substitution

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

There are no apparent substitutions for solving this integral, but the integral can be expressed as the sum of two separate integrals because this is a property of indefinite integrals.

The first integral can be solved with a simple u-substitution where 

The integral can be rewritten as:

To finally solve this, there is no other way to do so other than knowing the following: 

Finally, the answer must be expressed in terms of :

The second integral is a bit more complicated. It can be noted that the second integral resembles the following:

Specifically, . The second integral can be rewritten as:

With each separate integral found, the answers can be added to equal the original integral:

 

 

 

 

 

Example Question #661 : Finding Integrals

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

To solve this, you can use a u-substitution where

.

Now, the integrand can be completely rewritten in terms of u:

Before trying to further solve this integral in a more complicated way, by remembering what u equals, the integrand can be rewritten as:

The integral was taken by using the following formula:

The original problem was in terms of x. Therefore, the final answer is:

Example Question #80 : Solving Integrals By Substitution

Use trigonometric substitution to set up the given integral in terms of . Do not evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

The trig sub is used to redefine the integral in terms of a different variable, , to make its evaluation possible. 

For the given integral, following steps are important to redefine it using trigonometric substitution:

1) Find out, which trigonometric identity is best for given integral. In our case, the integrand is: . Therefore,   is an appropriate identity.

2) Redefine the bounds of integration:

3) Change  to . Differentiating  from part (1):

4) Rewrite the integral in terms of :

Example Question #81 : Solving Integrals By Substitution

Calculate the following integral: 

Possible Answers:

Correct answer:

Explanation:

Solve via integration by parts. Make the following substitutions:

    .

Plug in substitutions: .

Solve via integration by parts again. Make the following substitutions:

      .

Plug in substitutions: .

Simplify: .

Plug this integral back into our original equation:

Simplify: .

Add  to both sides of the equation: .

Factor  out of the right side of the equation: 

Divide both sides of the equation by 

Example Question #81 : Solving Integrals By Substitution

Calculate the following integral: 

Possible Answers:

Correct answer:

Explanation:

Factor out a sinx:

Apply Pythagorean identity to .

Making the following substitutions:   

Apply substitutions: 

Solve integral: 

Convert u back to x: 

Example Question #83 : Solving Integrals By Substitution

Calculate the following integral: 

Possible Answers:

Correct answer:

Explanation:

Make the following substitution:  .

Plug the substitution into the integrand: .

Factor out  from the denominator, and simplify: .

Apply Pythagorean identity to the denominator, and simplify:.

Solve integral:.

Use original substitution to solve for        

Plug value for  back into solution for integral: .

Example Question #82 : Solving Integrals By Substitution

Evaluate the following indefinite integral: 

Possible Answers:

Correct answer:

Explanation:

This integral, , is a classic integration by substation problem. This is indicated by the presence of a composite function in which both a function and its derivative are present. 

For this we need to let our variable  , substituting this into our integral we produce the following:

and by substituting back in for u we find our final answer to be:
.

Example Question #85 : Solving Integrals By Substitution

Evaluate the integral and use the sine function to make an appropriate trigonometric substitution. 

 

Possible Answers:

Correct answer:

Explanation:

                                                    (1)

 

To compute this integral we will use a trigonometric substitution. First we need to do some algebra to write the integrated in a more suitable form, 

 

                                               (2)

 

Now we can apply a trigonometric substitution. 

                                                   (3)

                                         (4)

 

 

Notice we've chosen this substitution so that the radicand in equation (2) will conveniently reduce as follows, 

 

  

 

So now we have  , we need to convert back to , use equation (3), 

 

 

Example Question #671 : Finding Integrals

Integrate

 



Possible Answers:

Correct answer:

Explanation:

                                                       (1)

1) Simplify with a substitution. 

It is often necessary to define a new variable , carefully chosen so that rewriting the integrand in terms of this new variable will make integration easier. In this case, the obvious variable to introduce will be defined by, 

                                                                      (2)

                                                                         (3)

 

Use equations (2) and (3) to rewrite (1).  

 

 

 2) Use integration by parts

To compute  use integration by parts. Ignore the constant out front for the moment, 

 

                                                 (4)

Define  and  and insert into the equation (4). 

  

 ,

 

 

 

 

                                               (5)

 

Let's factor the non-constant terms in equation (5), this will make the result easier to express when we convert back to 

 

We previously had a constant in front of the integral, 

 

 

Now we can write the integral terms of the original variable  by substituting equation (2) into the previous expression to obtain, 

 

 

 

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