Solve Tricky Integration: Compute Value of 1∫g(ln(t))/t dt

In summary, the conversation discusses the method of u-substitution to solve for the integral of g(ln(t))/t dt, given the value of 0\int1 g(t)dt=5. The suggestion is to also substitute the boundaries when using u-substitution.
  • #1
jj2443
10
0

Homework Statement



Suppose that the 0[tex]\int[/tex]1 g(t)dt=5.
Compute the value of 1[tex]\int[/tex]e g(ln(t))/t dt

Homework Equations


The Attempt at a Solution


I think that u substitution is the best way to solve.
If you set u=ln(t), then du=1/t dt which is in your integration. I do not know how to incorporate the given value of the integral of g(t) into the second integration, especially because the bounds of integration are different.

Thank you for any help!
 
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  • #2
When you u-substitute, substitute your boundaries as well. For example, when t = 1, what is ln(t)? Repeat with e.
 

FAQ: Solve Tricky Integration: Compute Value of 1∫g(ln(t))/t dt

1. How do I approach solving this tricky integration problem?

When dealing with tricky integration problems, it is important to start by identifying any potential substitution or manipulation techniques that can be applied. In this case, the presence of a natural logarithm function in the integrand suggests using u-substitution with u = ln(t). Then, using the formula for integration by parts, the problem can be solved.

2. What is the value of the integral 1∫g(ln(t))/t dt?

The exact value of this integral will depend on the function g(x) that is being integrated. However, once the substitution u = ln(t) is made, the integral can be rewritten as ∫g(u) du, which can then be evaluated using integration techniques such as the power rule or trigonometric substitution.

3. Can I use a calculator to solve this integral?

While a calculator or computer program may be able to provide a numerical approximation of the integral, it is important to understand the underlying concepts and techniques used in solving integration problems. These skills are crucial in more advanced mathematical and scientific fields.

4. Are there any other methods I can use to solve this integration problem?

There are many different methods that can be used to solve integration problems, including substitution, integration by parts, partial fractions, and trigonometric substitution. It is important to practice and become comfortable with these techniques in order to be able to approach a variety of integration problems.

5. Why is integration important in scientific research?

Integration is a crucial tool in many areas of scientific research, including physics, engineering, and economics. It allows us to find the area under a curve, which has many practical applications such as calculating volumes, finding the distance traveled by an object, and determining the probability of certain events. Integration also helps us to model and understand complex systems and phenomena in the natural world.

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