Evaluating Integral with a natural log

In summary, the formula for evaluating an integral with a natural log is ∫ln(x)dx = xln(x) - x + C. The process involves using integration by parts or substitution. Limits of integration can be used, but must be substituted into the formula. There are special cases, such as ∫ln(ax)dx and ∫ln(x^a)dx, where a is a constant. Tips for simplifying integrals with natural logs include algebraic manipulation and using substitution to identify patterns.
  • #1
XedLos
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Homework Statement


[tex]\int9s9^s ds[/tex]

Homework Equations


∫udv=uv-∫9^sds-∫vdu

The Attempt at a Solution


u=9s
du=9ds

dv=9^s ds
v=∫9^sds
=∫3^(2s) ds
=3^(2s)/[2ln(3)]
this is as far as i have gotten. Am i correct so far?
 
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  • #2
Yes, that is correct. You didn't really need to reduce to 3, that is exactly the same as
[tex]\int 9^s ds= 9^s/ln(9)[/tex]
 

FAQ: Evaluating Integral with a natural log

What is the formula for evaluating an integral with a natural log?

The general formula for evaluating an integral with a natural log is ∫ln(x)dx = xln(x) - x + C.

What is the process for evaluating an integral with a natural log?

The process for evaluating an integral with a natural log involves using integration by parts or substitution to convert it into a form that can be solved using the formula ∫ln(x)dx = xln(x) - x + C.

Can an integral with a natural log have limits of integration?

Yes, integrals with natural logs can have limits of integration. These limits must be substituted into the general formula for evaluating an integral with a natural log.

Are there any special cases for evaluating an integral with a natural log?

Yes, there are a few special cases for evaluating an integral with a natural log. These include integrals with the form ∫ln(ax)dx and ∫ln(x^a)dx, where a is a constant.

Are there any tips for simplifying integrals with natural logs?

One tip for simplifying integrals with natural logs is to use algebraic manipulation to combine the natural log terms. Another tip is to look for patterns and use substitution to simplify the integral.

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