Solve Logarithmic Integral int. (ln(ax+b))^2

Thread starter mikaela_clare
Start date Jul 9, 2010
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Integral Logarithmic

In summary, a logarithmic integral is an integral that involves the natural logarithm function and is used to calculate the area under the curve of the natural logarithm function. To solve such integrals, one can use the substitution method or integration by parts. The general formula for solving a logarithmic integral is given by ∫ln(ax+b)dx = (ax+b)(ln(ax+b) - 1) + C, which can be derived using integration by parts. A specific example of solving a logarithmic integral, int. (ln(ax+b))^2, involves using the substitution u = ln(ax+b) and the power rule. There are also special cases for solving logarithmic integrals, such as when the logarithmic function is

Jul 9, 2010

mikaela_clare

Does anyone know how to solve this integral?
int. (ln(ax+b))^2

Struggling!
Thanks :D

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Jul 9, 2010

losiu99

By parts (u = dv =log(ax + b) )

Last edited: Jul 9, 2010

Jun 4, 2011

soumyadipta

do it by parts
assume 1 as the second part and continue.
the logarithmic part will break down to fractions which can be easily integrated

1. What is a logarithmic integral?

A logarithmic integral is a special type of integral that involves the natural logarithm function. It is written as ∫ln(x)dx and is used to calculate the area under the curve of the natural logarithm function.

2. How do you solve an integral with a logarithmic function?

To solve an integral with a logarithmic function, you can use the substitution method or integration by parts. In the case of the integral int. (ln(ax+b))^2, you can use the substitution u = ln(ax+b) to simplify the problem.

3. What is the general formula for solving a logarithmic integral?

The general formula for solving a logarithmic integral is ∫ln(ax+b)dx = (ax+b)(ln(ax+b) - 1) + C. This formula can be derived using integration by parts.

4. Can you explain the steps to solve the integral int. (ln(ax+b))^2?

First, use the substitution u = ln(ax+b) to simplify the integral to ∫u^2du. Then, use the power rule to solve this integral, which results in (u^3)/3 + C. Finally, substitute back in u = ln(ax+b) to get the final answer of (ln(ax+b))^3/3 + C.

5. Are there any special cases for solving logarithmic integrals?

Yes, there are a few special cases for solving logarithmic integrals. For example, if the logarithmic function is in the denominator, you can use the substitution u = ln(x) to simplify the integral. Additionally, if the logarithmic function is raised to a fraction or a negative power, you can use the substitution u = ln(ax+b) and some algebraic manipulation to simplify the integral.