Answer

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**Hint:**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.

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