Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #171 : Integral Expressions

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate this integral, it is easier to split it into two integrals:

For the first integral, the following substitution must be made:

Rewriting the integral and integrating, we get:

The following rule was used to integrate:

Replacing u with the original term, we get

Next, we evaluate the second integral:

which was found using the following rule:

Combining the results - and the constants of integration - we get

 

Example Question #171 : Integral Expressions

Set up, but don't evaluate an integral expression which will find the area of the region bound by the lines , the x-axis, and the function v(x)

Possible Answers:

Correct answer:

Explanation:

Set up, but don't evaluate an integral expression which will find the area of the region bound by the lines , the x-axis, and the function v(x)

First, we need the correct limits of integration. We want 5 as the lower limit and 25 as the upper limit.

Next, add in our function and the dx and we are all set!

Example Question #173 : Integral Expressions

Set up, but do not evaluate, an integral expression for the area of the region bound by the y-axis, h(t), and the lines 

Possible Answers:

Correct answer:

Explanation:

Set up, but do not evaluate, an integral expression for the area of the region bound by the y-axis, h(t), and the lines 

To set up a definite integral, we first need our limits of integration:

In this case, they will be 5 and 6, because those are the lines which are the boundaries of our region.

Finally, we want to include our function

Don't forget the dt, so that we know which variable we would integrate with respect to.

 

Example Question #171 : 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 #175 : Integral Expressions

Find the indefenite integral 

Possible Answers:

Correct answer:

Explanation:

We need to use substitution to solve.  Let

Differentiating, gives

Solving for dx gives

Plugging into the original integral gives

Simplifying gives

Plugging back in for u gives

Example Question #172 : How To Find Integral Expressions

Find the indefenite integral 

Possible Answers:

Correct answer:

Explanation:

We need to use substitution to solve.  Let

Differentiating gives

Solving for dx, so that we can plug back in, gives

Plugging in gives

Simplifying gives

Integrating gives

Plugging u back in gives

Example Question #171 : How To Find Integral Expressions

Find the indefenite integral 

Possible Answers:

Correct answer:

Explanation:

We need to use integration by parts, which says

Let

Differentiating u and integrating dv, gives

Plugging into the equation, gives

Now, we can integrate the last term to get

Example Question #173 : How To Find Integral Expressions

Find the indefenite integral 

Possible Answers:

Correct answer:

Explanation:

We need to use integration by parts, which says

Let

Differentiating u and integrating dv, gives

Plugging into the equation, gives

Now, we can integrate the last term to get

Example Question #174 : How To Find Integral Expressions

Solve the integral:

Possible Answers:

Correct answer:

Explanation:

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

The derivative was found using the following rule:

Now, rewrite the integral and integrate:

The integral was found using the rule

Finally, replace u with the original term:

.

 

Example Question #171 : Integral Expressions

A very important physics formula is called the continuity equation. We will only consider the 1-dimensional continuity equation for now. 

In the continuity equation, we're given that:

, where  is a function of  and .

Rewrite the right side of the equation in integral form. 

Possible Answers:

Correct answer:

Explanation:

Recall that the fundamental theorem of calculus states that:

Using this knowledge we write the right side as:

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