# How do you integrate ln(5x+3)?

May 19, 2015

Edit: I misread the question- I didn't integrate, I differentiated.

$\frac{d}{\mathrm{dx}} \ln \left(5 x + 2\right) = \frac{5}{5 x + 3}$

Solution

You would do this using the chain rule.

The chain rule, in words, basically just means:

the derivative of the outer function(leaving the inner function alone, or treating it as a single variable) X the derivative of the inner function

Here, seeing the "outer" and "inner" functions is pretty straightforward.

We have $\ln \left(5 x + 3\right)$. Just by looking at it, you can see that $5 x + 3$ is "inside" the $\ln$, making it the inner function.

Now we can do the chain rule. We know that the derivative of $\ln \left(u\right)$, for example, is just $\frac{1}{u}$. Well, the derivative of $\ln \left(5 x + 3\right)$ (while leaving the inner function alone, or treating is as "u"!) is $\frac{1}{5 x + 3}$. But now, to complete the chain rule, we have to multiply by the derivative of the inner function- The derivative of $5 x + 3$ is simply $5$.

So the final answer is:

$\frac{d}{\mathrm{dx}} \ln \left(5 x + 2\right) = \frac{1}{5 x + 3} \cdot 5 = \frac{5}{5 x + 3}$

May 19, 2015

$\int \ln \left(5 x + 3\right) \mathrm{dx}$.

Let $w = 5 x + 3$, so that $\mathrm{dw} = 5 \mathrm{dx}$ and the integral becomes:

$\frac{1}{5} \int \ln w \mathrm{dw}$

Integrate by parts: $u = \ln w$ and $\mathrm{dv} = \mathrm{dw}$ this makes:

$\mathrm{du} = \frac{1}{w} \mathrm{dw}$ and $v = w$, using the formula for integral by parts:

$\frac{1}{5} \int \ln w \mathrm{dw} = \frac{1}{5} \left[w \ln w - \int w \cdot \frac{1}{w} \mathrm{dw}\right]$

$\textcolor{w h i t e}{\text{ssssssssssss}}$ $= \frac{1}{5} \left[w \ln w - \int \mathrm{dw}\right]$

$\textcolor{w h i t e}{\text{ssssssssssss}}$ $= \frac{1}{5} \left[w \ln w - w\right] + C$

Therefore,

$\int \ln \left(5 x + 3\right) \mathrm{dx} = \frac{1}{5} \left[\left(5 x + 3\right) \ln \left(5 x + 3\right) - \left(5 x + 3\right)\right] + C$.

$\textcolor{w h i t e}{\text{sssssssssssssss}}$ $= \frac{5 x + 3}{5} \left(\ln \left(5 x + 3\right) - 1\right) + C$