Integral of ln

[Jason76]

What is the integral of \(\displaystyle \ln\)? Example \(\displaystyle \int \ln(3x + 5) = ?\) I know that the integral of a 1 over an expression is a \(\displaystyle \ln\) expression. For instance
\(\displaystyle \int \dfrac{1}{3x + 5} dx = \ln(3x + 5) + C\)

 

[daon2]

I believe you are in a first Calculus course. The technique for integrating a logarithm is not taught until (in my experience) a second course in calculus. The technique is called integration by parts. See: http://en.wikipedia.org/wiki/Integration_by_parts

The answer is \(\displaystyle \int \ln(x) dx = x\ln(x)-x + C = x(\ln(x)-1) + C\)

 

[Jason76]

I believe you are in a first Calculus course. The technique for integrating a logarithm is not taught until (in my experience) a second course in calculus. The technique is called integration by parts. See: http://en.wikipedia.org/wiki/Integration_by_parts

The answer is \(\displaystyle \int \ln(x) dx = x\ln(x)-x + C = x(\ln(x)-1) + C\)

Thanks, so that's the general formula? What about \(\displaystyle \int \ln(4x - 3) dx\)?

 

[MarkFL]

In order to apply the formula give by daon2, you will want to use a u-substitution first:

\(\displaystyle u=4x-3\,\therefore\,du=4\,dx\)

and then your integral becomes:

\(\displaystyle \displaystyle \frac{1}{4}\int\ln(u)\,du\)

Now you can apply the formula, then back-substitute for u.

 

[Deleted member 4993]

What is the integral of \(\displaystyle \ln\)? Example \(\displaystyle \int \ln(3x + 5) = ?\) I know that the integral of a 1 over an expression is a \(\displaystyle \ln\) expression. For instance
\(\displaystyle \int \dfrac{1}{3x + 5} dx = \ln(3x + 5) + C\).......... Incorrect

\(\displaystyle \int \dfrac{1}{3x + 5} dx = \frac{1}{3}\ln(|3x + 5|) + C\)

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