# Fundamental Theorem Of Calculus problems help

• raepal
In summary: There is a simple substitution you can make, and then you should be able to evaluate the integral directly.In summary, the conversation discusses three problems which involve finding the derivative using the Fundamental Theorem of Calculus. The first problem involves finding the derivative of g(x) using the limits [8x to 4x] and [4x to 8x]. The second problem, h(x), involves a definite integral with limits from sin(x) to -3. The third problem, F(x), involves a definite integral with limits from 1 to the square root of 3. The conversation also discusses the use of substitution to simplify the integration process.
raepal
Fundamental Theorem Of Calculus problems help!

## Homework Statement

A))))

Find the derivative of
g(x)=∫[8x to 4x] (u+7)/(u-4) dx

B)))
Use part I of the Fundamental Theorem of Calculus to find the derivative of
h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt

C)))
F(x) = ∫[ 1 to √3] s^3/(3+5s^4) dx

## The Attempt at a Solution

I tried to do
F(b)b' - F(a)a'
but I am not confident with my answer.

raepal said:

## Homework Statement

A))))

Find the derivative of
g(x)=∫[8x to 4x] (u+7)/(u-4) dx

B)))
Use part I of the Fundamental Theorem of Calculus to find the derivative of
h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt

C)))
F(x) = ∫[ 1 to √3] s^3/(3+5s^4) dx

## The Attempt at a Solution

I tried to do
F(b)b' - F(a)a'
but I am not confident with my answer.
What does part I of the Fundamental Thm. of Calculus say?

raepal said:

## Homework Statement

A))))

Find the derivative of
g(x)=∫[8x to 4x] (u+7)/(u-4) dx

B)))
Use part I of the Fundamental Theorem of Calculus to find the derivative of
h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt

C)))
F(x) = ∫[ 1 to √3] s^3/(3+5s^4) dx

## The Attempt at a Solution

I tried to do
F(b)b' - F(a)a'
but I am not confident with my answer.

Are you sure you mean g(x) = ∫[8x to 4x] (u+7)/(u-4) dx in A)? As written, this means
$$g(x) = \int_{8x}^{4x} \frac{u+7}{u-4} \, du .$$ (Where you wrote 'dx' I assume you mean 'du'.) The point is: what are the limits, and where they go? If you actually want $$g(x) = \int_{4x}^{8x} \frac{u+7}{u-4} \, du,$$ you would need to write
∫[4x to 8x] (u+7)/(u-4) du. The answers for g'(x) are different in these two cases. Which one do you really mean?

Ray Vickson said:
Are you sure you mean g(x) = ∫[8x to 4x] (u+7)/(u-4) dx in A)? As written, this means
$$g(x) = \int_{8x}^{4x} \frac{u+7}{u-4} \, du .$$ (Where you wrote 'dx' I assume you mean 'du'.) The point is: what are the limits, and where they go? If you actually want $$g(x) = \int_{4x}^{8x} \frac{u+7}{u-4} \, du,$$ you would need to write
∫[4x to 8x] (u+7)/(u-4) du. The answers for g'(x) are different in these two cases. Which one do you really mean?

It's the second one that you stated.
here is it:

Is this problem B?
You wrote "h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt"

From your response to Ray, I think this is the function.

$$h(x) = \int_{-3}^{sin(x)}(cos(t^3) + t)dt$$

Can you work the problem below? This is a little easier, and if you can work it, the one above is only a little harder.
$$h(x) = \int_{-3}^x(cos(t^3) + t)dt$$

Let me ask again, what does the first part of the Fundamental Thm. of Calculus say?For problem C you wrote "F(x) = ∫[ 1 to √3] s^3/(3+5s^4) dx"

I can only guess at what you meant, which might be this:
$$F(x) = \int_1^{\sqrt{3}} \frac{s^3}{3 + 5s^4} ds$$

If this is the problem, it's very easy.

## What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a fundamental concept in calculus that states the relationship between differentiation and integration. It essentially states that integration is the inverse operation of differentiation.

## What are some common types of Fundamental Theorem of Calculus problems?

Some common types of Fundamental Theorem of Calculus problems include finding the derivative of an integral, evaluating definite integrals using the Fundamental Theorem of Calculus, and using the Fundamental Theorem of Calculus to find the area under a curve.

## How do I know when to use the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is typically used when dealing with problems involving the relationship between differentiation and integration. This includes problems where you need to find derivatives of integrals or evaluate definite integrals.

## What are the steps for solving a Fundamental Theorem of Calculus problem?

The steps for solving a Fundamental Theorem of Calculus problem involve first identifying the type of problem you are dealing with, using the appropriate formula or rule, and then simplifying the problem until you can find the solution. It is important to also pay attention to any given constants or limits in the problem.

## What are some tips for successfully solving Fundamental Theorem of Calculus problems?

Some tips for successfully solving Fundamental Theorem of Calculus problems include practicing with a variety of problems, understanding the relationship between differentiation and integration, and carefully reading and following the given instructions or limits in the problem.

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