Indefinite Integration with Logarithms and Substitution

In summary, the conversation is about a calculus problem involving finding the indefinite integral of 1/(x ln(x^3)). The person has tried using substitution and applying log properties, but is having difficulty solving the problem. The other person provides a hint that ln(x^3)=3ln(x) and suggests using 1/3*ln|ln(x)| as the solution.
  • #1
Brilliant
38
0
Hi, I missed a few days of my calculus class. I've managed to figure out how to use substitution to solve an indefinite integral, and can apply the log properties to some extent. I just can't figure out this problem.

Homework Statement


Find the indefinite integral:
[tex]\int{\frac{1}{x ln(x^3)}}dx[/tex]

2. The attempt at a solution
Well, since d/dx ln(x) is u'/u I know something is kinda wack with the bottom. I first tried to substitute with u=x^3, but then du is 3x^2 and there is only x on the bottom not x squared. I then thought it might be backwards, since the x on bottom is like x^-1 and it would have an integral with a natural log in it, but that wasn't really working out either.

I'm pretty stumped, I've attempted it several times.

Thanks for your help!
 
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  • #2
Hint: ln(x^3)=3ln(x).
 
  • #3
I hate it when that happens. so 1/3*ln|ln(x)|

Thanks grief!
 

Related to Indefinite Integration with Logarithms and Substitution

1. What is indefinite integration with logarithms and substitution?

Indefinite integration with logarithms and substitution is a mathematical technique used to find the antiderivative of a function that involves logarithmic and other polynomial terms. It involves using a change of variables or substitution to transform the integral into a simpler form that can be easily evaluated.

2. Why is substitution used in indefinite integration with logarithms?

Substitution is used in indefinite integration with logarithms because it allows us to replace complex expressions with simpler ones and make the integral easier to solve. It also helps in reducing the number of terms in the integral, making it easier to evaluate.

3. What is the process for indefinite integration with logarithms and substitution?

The process for indefinite integration with logarithms and substitution involves identifying the appropriate substitution, substituting for the variable, simplifying the expression, integrating, and finally substituting back the original variable. The substitution is usually chosen to cancel out terms and make the integral easier to evaluate.

4. Can indefinite integration with logarithms and substitution be used for all functions?

No, indefinite integration with logarithms and substitution cannot be used for all functions. It is most effective for functions that involve logarithmic and polynomial terms. For other types of functions, different integration techniques may be more appropriate.

5. What are some common mistakes to avoid in indefinite integration with logarithms and substitution?

Some common mistakes to avoid in indefinite integration with logarithms and substitution include choosing the wrong substitution, forgetting to substitute back the original variable, and making algebraic errors while simplifying the expression. It is important to carefully follow the steps and check the final answer for accuracy.

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