I'd like some feedback on my approach to solving this integral

In summary, the conversation discussed the approach to solving an indefinite integral of an exponential function using the u-substitution method. The question was whether this approach was efficient or if there was a faster way to solve it. The image provided showed the steps taken to solve the integral. It was then suggested that a simpler method could be used by noting that a logarithm of the variable can be rewritten as x. Additionally, it was reminded to always include the constant of integration when solving integrals.
  • #1
tomvidm
3
5
1. I gave myself the task of solving the indefinite integral of an exponential function whose base is any real valued constant and whose power is a logarithm of the variable I am integrating over. Now my question is not how to solve it, but rather, whether or not my approach was efficient. I've just started practicing doing the u-substitutions, so my approach was centered around that, but it also involved integration by parts. Below is an image of my paper containing all the information I think is needed.

6AjLzpHl.jpg


Was my choice of technique good or could I have solved this integral in some other manner?
(Also, I removed the points 2) and 3) which comes with every thread here. I think the image speaks for itself.)
 
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  • #2
There is a faster way. Note that [itex] a^{lnx}=(e^{lna})^{lnx}=(e^{lnx})^{lna}=x^{lna} [/itex], which is very easy to integrate. Your answer can be simplified by noting that [itex] e^{lnx}=x [/itex] as long as x>0, and the two answers will be identical.

Edit: This is a minor point, but never forget your constant of integration!
 
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FAQ: I'd like some feedback on my approach to solving this integral

1. How do I approach solving integrals?

There are multiple approaches to solving integrals, but a commonly used method is to first identify the type of integral (such as trigonometric, logarithmic, or polynomial) and then use appropriate techniques such as substitution, integration by parts, or partial fractions.

2. What should I do if I get stuck on a certain step while solving an integral?

If you are struggling with a particular step, you can try looking for online resources or consulting with a tutor or classmate. It may also be helpful to go back and review the basic rules and techniques of integration.

3. How do I know if my approach to solving an integral is correct?

You can verify your answer by taking the derivative of the result and seeing if it matches the original integrand. Additionally, you can check your work using online integral calculators or by comparing your solution to the answer in a textbook or solution manual.

4. Are there any shortcuts or tricks for solving integrals?

Yes, there are some common tricks and shortcuts that can be used for certain types of integrals. For example, the substitution method can often simplify integrals involving trigonometric functions, and integration by parts can be helpful for products of functions.

5. How can I improve my skills in solving integrals?

The best way to improve your skills in solving integrals is through practice. Make sure to understand the basic rules and techniques, and then work on a variety of problems to become familiar with different types of integrals and how to approach them.

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