Solving Integrals with e: Homework Equations & Solutions

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SUMMARY

The integral of the function \( e^{u+1} \) from -1 to 1 can be solved by recognizing that \( e^{1} \) is a constant. The correct approach involves rewriting the integral as \( \int_{-1}^{1} e^{u} e^{1} \, du \). The integral of \( e^{u} \) is \( e^{u} \), and since \( e^{1} \) is a constant, it factors out of the integral. The final result is \( e^{2} - 1 \), confirming that the integration process was correctly applied once the constant was identified.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with exponential functions, specifically \( e^{x} \)
  • Knowledge of integration rules, including the constant factor rule
  • Ability to manipulate and rewrite integrals
NEXT STEPS
  • Study the properties of exponential functions, focusing on \( e^{x} \)
  • Learn about integration techniques for functions involving constants
  • Practice solving definite integrals with various exponential functions
  • Explore the application of the constant factor rule in integration
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Students studying calculus, particularly those struggling with integrals involving exponential functions, as well as educators looking for examples to illustrate integration techniques.

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

I thought maybe e^{1} would just integrate like a normal function giving 1/2e^{2} 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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This is a commonly known rule:

\int af(x) dx = a\int f(x) dx

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

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