# Integrating Equations with Exponents: A Challenge

• Shambles
In summary, the conversation is about using substitution in an integral involving u = 1+tant and du = sec^2(t) dt. The individual is trying to figure out how to properly use substitution with an exponent and is seeking advice on how to solve the problem. They eventually realize that they can simply substitute u for (1 + tan t) and du for sec^2(t) dt, resulting in an integral of u^3 with limits of integration from 1 to 2.
Shambles

## Homework Equations

u = 1+tant
du = sec^2(t) dt
dt = du / sec^2(t)

## The Attempt at a Solution

It seems like I should be using substitution in the equation, however the exponent is messing things up for me. I recall from derivatives how they interact with the chain rule, but am not sure how this would work backwards in integration. Something like,

I(u^3)(sec^2(t)) = (u^4/4)((sec^2(t)) (tan(t))

Except I haven't gotten rid of the t variable and now have t and u. Any points are welcome.

Why don't you just substitute u for (1 + tan t) and du for sec^2(t) dt (and take care of the limits of integration, of course)?

Ah I see how when I change the limits of integration it removes the nasty sec^2(t) so all I'm left with is the integral of u^3 with u going from 1 to 2. Thanks.

## 1. What is the purpose of integrating equations with exponents?

The purpose of integrating equations with exponents is to solve for the area under the curve of a function. This is useful in many fields of science, including physics, engineering, and economics.

## 2. What are the steps involved in integrating equations with exponents?

The first step is to identify the exponent and determine if it is a power rule, exponential rule, or logarithmic rule. Then, rewrite the equation using the appropriate rule. Next, integrate the equation using the power rule, exponential rule, or logarithmic rule. Finally, solve for the constant of integration.

## 3. What are some common challenges when integrating equations with exponents?

Some common challenges include correctly identifying the exponent and choosing the correct integration method. Additionally, the constant of integration can be a source of confusion if not properly accounted for.

## 4. How can I practice and improve my skills in integrating equations with exponents?

The best way to practice and improve your skills is to solve many different types of problems. You can find practice problems online or in textbooks, and it can also be helpful to work with a tutor or study group to discuss your thought process and any difficulties you encounter.

## 5. How is integrating equations with exponents used in real-world applications?

Integrating equations with exponents is used to solve a variety of problems in fields such as physics, engineering, and economics. For example, it can be used to calculate the work done by a variable force, the growth of a population, or the value of a financial investment over time.

• Calculus and Beyond Homework Help
Replies
8
Views
906
• Calculus and Beyond Homework Help
Replies
22
Views
2K
• Calculus and Beyond Homework Help
Replies
15
Views
969
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
611
• Calculus and Beyond Homework Help
Replies
3
Views
745
• Calculus and Beyond Homework Help
Replies
9
Views
275
• Calculus and Beyond Homework Help
Replies
21
Views
1K