Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #151 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

Before integrating, I would multiply the binomials (using FOIL) together to get a polynomial: .  Now, integrate. Remember, when integrating, raise the exponent by 1 and then put that result on the denominator as well. You then get: . Simplify and add a "C" at the end because it is an indefinite integral: .

Example Question #152 : How To Find Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this expression, first take the 4 outside of the integral expression: . Recall that when you are integrating , the result is . Then, multiply by 4 and tack on a "C" because it is an indefinite integral: .

Example Question #153 : Integral Expressions

Possible Answers:

Correct answer:

Explanation:

To integrate this expression, raise the exponent by 1 and then put that result on the denominator (while multiplying that expression by any coefficients). Therefore, when you integrate, you get: . Then, simplify and add "C" because it is an indefinite integral so that your answer is: .

Example Question #153 : How To Find Integral Expressions

Evaluate the following Indefinte Integral:

Possible Answers:

Correct answer:

Explanation:

For this problem one must remember the following rule for Integrals: 

Once applying this rule we end with the following two integrals:

 

Now apply our basic rules for Trigonometric Integrals we find the following:

Note: a "+C" is nessecary as this integral is indefinite, that is, we are not plugging in any limits of integration.

Example Question #154 : How To Find Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

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

Next, we rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with our original term:

Example Question #155 : 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:

Example Question #157 : Integral Expressions

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 #158 : Integral Expressions

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 #159 : Integral Expressions

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 #160 : Integral Expressions

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:

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