Question

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

Answer 1

We will use the ILATE rule and take the given first function and take 1 (constant) as the second function. Now, we will just use the normal formulas to find its integration.

Complete step by step answer:
We are given that we are required to find the integration of ln (5x + 3).
This means that we are required to find $\int {\ln (5x + 3)dx}$.
We can definitely write it as $\int {\ln (5x + 3).1dx}$.
Using the ILATE rule, we see that the first function will be ln (5x + 3) and the second function will be 1.

Now, we will use the formulas mentioned as follows:-
\Rightarrow \int {f(x)g(x)dx = f(x)} \int {g(x)dx - \int {\left\\{ {\dfrac{d}{{dx}}f(x)\int {g(x)dx} } \right\\}dx} } where f (x) is the first function and g (x) is the second function.
Putting f (x) as ln (5x + 3) and g (x) as 1, we will then obtain:-
\Rightarrow \int {\ln (5x + 3)dx = \ln (5x + 3)} \int {dx - \int {\left\\{ {\dfrac{d}{{dx}}\ln (5x + 3)\int {dx} } \right\\}dx} }

Now, since we know that the integration of dx is always x, we will then obtain the following equation:-
\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - \int {\left\\{ {\dfrac{d}{{dx}}\ln (5x + 3)} \right\\}xdx}
Now, we also know that $\dfrac{d}{{dx}}[\ln (ax + b)] = \dfrac{a}{{ax + b}}$. Using this, we will then obtain the following equation:-
$\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - \int {\dfrac{{5x}}{{5x + 3}}dx}$

Now, we can also write the above equation as:-
$\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - \int {\dfrac{{5x + 3 - 3}}{{5x + 3}}dx}$
Now, we can also write the above equation as:-
$\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - \int {\dfrac{{5x + 3}}{{5x + 3}}dx} + \int {\dfrac{3}{{5x + 3}}dx}$

Now, we also know that $\int {\dfrac{1}{{ax + b}}dx = \dfrac{{\ln (ax + b)}}{a}}$. Using this, we will then obtain the following equation:-
$\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - \int {dx} + \dfrac{{3\ln (5x + 3)}}{5}$
Simplifying the integration on the right hand side further, we will then obtain the following equation:-
$\Rightarrow \int {\ln (5x + 3)dx = x\ln (5x + 3)} - x + \dfrac{{3\ln (5x + 3)}}{5} + C$
$\Rightarrow \int {\ln (5x + 3)dx = \dfrac{{5x\ln (5x + 3) - 5x + 3\ln (5x + 3)}}{5}} + C$
$\Rightarrow \int {\ln (5x + 3)dx = \dfrac{{(5x + 3)\ln (5x + 3) - 5x}}{5}} + C$

Note: For example:- $\int {af(x)dx} = a\int {f(x)dx}$.
make sure your using the ILATE rule which is as follows:-
I stands for Inverse, L stands for logarithmic, A stands for algebraic, T stands for trigonometric and E stands for exponential. We take the first function according to this preference only.
Like in the above solution, we termed logarithmic as first function and algebraic as second function.