Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #161 : Writing Equations

Find the area under the curve  between  and .

Possible Answers:

Correct answer:

Explanation:

In order to find the area underneath a curve for a specific range of values, we need to set up a definite integral expression. To do this, we plug our specific information into the following skeleton:

We plug in the equation of the curve in for , the smaller of our x-values for , and the larger of our x-values for . For this problem, our set-up looks like this:

Next, we integrate our expression, ignoring our  and  values for now. We get:

Now, to find the area under the part of the curve we're looking for, we plug our  and  values into our integrated expression and find the difference, using the following skeleton:

In this problem, this looks like:

Which plugged into our integrated expression is:

 

With our final answer being:

Example Question #162 : Writing Equations

Possible Answers:

Correct answer:

Explanation:

First, integrate the expression. Remember, when integrating, raise the exponent by 1 and then put that result on the denominator as well: . Then, evaluate that expression by plugging in 2 and then subtract from that the result from plugging in 0: .

Example Question #161 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

The first step I'd recommend is to chop up the expression into 3 terms since we have one denominator: . Now, integrate each term, remembering to add one to the exponent and then putting that result in the denominator as well: Tack on a "+C" at the end because it is an indefinite integral.

Example Question #161 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

The first step I'd recommend is to chop up the expression into 3 terms since we have one denominator: . Now, integrate each term, remembering to add one to the exponent and then putting that result in the denominator as well: Tack on a "+C" at the end because it is an indefinite integral.

Example Question #162 : How To Find Integral Expressions

Evaluate the following integral:

 

Possible Answers:

Correct answer:

Explanation:

The integral is equal to 

and was found using the following rules:

Note that all of the constants of integration added up to a single C.

Example Question #163 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we can split the integral into the sum of two integrals:

The first integral can be solved using a substitution:

and the integral rewritten in terms of u is

The integral was solved using the identical rule.

Now, we replace u with our original term:

.

Solving the second integral, we get

using the following rule:

Summing the results, we get

Example Question #164 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

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

The derivative was found using the following rule:

Next, we rewrite the integral and integrate:

The integral was found using the following rule:

Finally, replace u with our original x term:

Example Question #165 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we can split the integral into three integrals, for ease of making substitutions later:

To integrate the first integral, we must make the following substitution:

Next, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with the original term:

The result of all three integrals, added together, is

and the other two integrals were integrated using the following rules:

,

Example Question #166 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

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

The derivative was found using the following rule:

Next, rewrite the integral in terms of u and integrate:

The integral was found using the following rule:

Finally, replace u with our original x term:

Example Question #167 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

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

The derivative was found using the following rule:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with our original term:

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