Find the Indefinite Integral Using Substitution: e^2x and (1+e^2x)^3

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In summary, a basic integral question is a mathematical problem that involves finding the area under a curve. To solve it, one needs to determine the limits of integration and use various techniques such as the Fundamental Theorem of Calculus. These questions have real-world applications, but common mistakes can include forgetting to include the constant of integration. Calculators can be used to solve basic integral questions, but a good understanding of the concepts is still important.
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undrcvrbro
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Homework Statement


Find the indefinite integral using substitution:
[tex]\int[/tex]e^2x(1+e^2x)^3 dx

Homework Equations





The Attempt at a Solution


I'm not sure how to start. What do I substitute? Any suggestions to get me started will be greatly appreciated.
 
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  • #2
put u=e^2x====>du=2e^2xdx
 
  • #3
Better to put u=(1+e^(2x)).
 
  • #4
Dick said:
Better to put u=(1+e^(2x)).

Yes, thanks dick. Thats what I ended up doing. I had just learned this stuff today, so I think msa's approach may have been a bit over my head.
 

1. What is a basic integral question?

A basic integral question is a mathematical problem that involves finding the area under a curve. It is a fundamental concept in calculus and is used to solve a variety of real-world problems.

2. How do I solve a basic integral question?

To solve a basic integral question, you need to first determine the limits of integration and the function that represents the curve. Then, you can use various techniques such as the Fundamental Theorem of Calculus, integration by substitution, or integration by parts to find the solution.

3. What are the applications of basic integral questions?

Basic integral questions have many real-world applications, such as calculating the volume of irregularly shaped objects, finding the distance traveled by an object with varying speed, and determining the average value of a function over an interval.

4. Are there any common mistakes when solving basic integral questions?

Yes, some common mistakes when solving basic integral questions include forgetting to include the constant of integration, using the wrong limits of integration, and making errors in algebraic manipulation. It is important to carefully check your work to avoid these mistakes.

5. Can I use a calculator to solve basic integral questions?

Yes, most calculators have a built-in integral function that can be used to solve basic integral questions. However, it is still important to have a good understanding of the concepts and techniques involved in solving integrals in order to use the calculator effectively.

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