Solving Integrals with e: Homework Equations & Solutions

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In summary, the conversation discusses the struggle with integrating equations involving e, specifically the equation \begin{equation} \int_{-1}^{1} e^{u+1} \end{equation}. The attempt at a solution involves rewriting the equation as \begin{equation}\int_{-1}^{1} e^{u} e^{1}\end{equation} and using the rule \begin{equation} \int af(x) dx = a\int f(x) dx \end{equation} for constant a. The equation is finally solved by realizing that \begin{equation} e^{1} \end{equation} is a constant.
  • #1
dlthompson81
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Homework Statement


\begin{equation} \int_{-1}^{1} e^{u+1} \end{equation}

Homework Equations




The Attempt at a Solution



I really seem to struggle with any problems with e in them. I think I may have missed some of the basic rules or something, but I can't seem to find what I missed.

My guess on this one would be to rewrite the equation into:

\begin{equation}\int_{-1}^{1} e^{u} e^{1}\end{equation}

I know that the integral of [itex]e^{u}[/itex] is [itex]e^{u}[/itex] but I don't know how to integrate [itex]e^{1}[/itex]. I'm not even sure if I rewrote the problem correctly. I know that the answer is [itex]e^{2}-1[/itex] but I can't seem to figure out how to get there.

I thought maybe [itex]e^{1}[/itex] would just integrate like a normal function giving [itex]1/2e^{2}[/itex] but I couldn't get it to work out with that either.

I'm totally lost with these e functions. What am I doing wrong?
 
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  • #2
This is a commonly known rule:

[tex]\int af(x) dx = a\int f(x) dx[/tex]

for constant a. Knowing this now...
 
  • #3
I don't understand how that rule applies exactly.
 
  • #4
Think more about it. Is e1 a constant?
 
  • #5
Ok. I got it now. I wasn't thinking of [itex]e^{1}[/itex] as a constant. Thanks for the help.
 

FAQ: Solving Integrals with e: Homework Equations & Solutions

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value or quantity of something, such as the distance traveled by an object or the amount of fluid flowing through a pipe.

How do you solve integrals with e?

To solve integrals with e, you can use the substitution method or the integration by parts method. In the substitution method, you replace the variable in the integral with a new variable that can be solved for using the natural logarithm. In the integration by parts method, you break down the integral into two parts and use the formula ∫u·dv = uv - ∫v·du to solve it.

What are some common applications of integrals with e?

Integrals with e are commonly used in physics and engineering to solve problems related to motion, force, and energy. They are also used in economics and finance to calculate growth rates and compound interest. In chemistry, integrals with e can be used to model the rate of reactions and determine the concentration of substances over time.

What are some tips for solving integrals with e?

Here are some tips for solving integrals with e:

  • Always check if the integral is in its simplest form before attempting to solve it.
  • Try using substitution or integration by parts if the integral is not in its simplest form.
  • Remember the properties of e, such as e^x * e^y = e^(x+y) and (e^x)' = e^x.
  • If the integral is in the form of e^x * f'(x), try using integration by parts with u = e^x and dv = f'(x).
  • Practice and familiarize yourself with different types of integrals with e to improve your problem-solving skills.

Are there any common mistakes to avoid when solving integrals with e?

Yes, here are some common mistakes to avoid when solving integrals with e:

  • Forgetting to include the constant of integration when solving indefinite integrals.
  • Incorrectly applying the properties of e, such as forgetting to add or multiply the exponents.
  • Not checking the limits of integration when solving definite integrals.
  • Confusing e with other variables, such as x or y, when using the substitution method.
  • Not simplifying the integral before attempting to solve it.

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