Question

How do you integrate $({x^3})\ln (x)dx$?

Answer 1

Hint:
According to the question given in the question we have to integrate $({x^3})\ln (x)dx$. So, first of all to integrate the given logarithmic function as given in the question we have to use the formula to find the integration by part method which is as mentioned below:

Formula used:
\[ \int {uvdx = } v\int {udx - \int {\left( {\dfrac{{dv}}{{dx}}\int {udx} } \right)dx............(A)} } \]
Where, u is the first term and v is the second term hence we choose our first and the second terms.
Now, to find the differentiation of the second we have to use the formula to find the differentiation as mentioned below:
$ \dfrac{{d\ln x}}{{dx}} = \dfrac{1}{x}.................(B)$
Now, to find the integration of the first term we have to use the formula to find the integration which is as mentioned below:
$ \int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c...........(C)} $
Where, c is the constant term.
Now, we just have to solve the obtained logarithmic expression to obtain the required integration of the given integral.

Complete step by step solution:
Step 1: First of all to integrate the given logarithmic function as given in the question we have to use the formula to find the integration by part method which is as mentioned in the solution hint. Hence,
$ \Rightarrow v = \ln x$and,
$ \Rightarrow u = {x^3}$
Step 2: Now, we have to substitute all the values in the formula (A) as mentioned in the solution hint to obtain the required integration. Hence,
$ \Rightarrow \ln x\int {{x^3}dx - \int {\left( {\dfrac{{d\ln x}}{{dx}}\int {{x^3}dx} } \right)} } dx$
Step 3: Now, to find the differentiation of the second we have to use the formula (B) to find the differentiation as mentioned in the solution hint. Hence,
$ \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \int {\left( {\dfrac{{{x^4}}}{4}\int {\ln xdx} } \right)dx} $
Step 4: Now, to find the integration of the first term we have to use the formula (C) to find the integration which is as mentioned in the solution hint. Hence,
$ \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \int {\left( {\dfrac{{{x^4}}}{4}\dfrac{1}{x}} \right)dx} $
On eliminating x in the integration as obtained just above,
$ \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \int {\left( {\dfrac{{{x^3}}}{4}} \right)dx} $
On taking the constant term outside the integration,
$ \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \dfrac{1}{4}\int {{x^3}dx} $
Step 5: Now, to solve the integration of the function as obtained in the solution step 4 we have to use the formula (C) which is as mentioned in the solution hint. Hence,
\[
   \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \dfrac{1}{4}\dfrac{{{x^4}}}{4} \\
   \Rightarrow \ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \dfrac{{{x^4}}}{{16}} \\
 \]

Hence, with the help of the formulas (A), (B) and (C) we have determined the required integration of the given integration which is \[\ln x\left( {\dfrac{{{x^4}}}{4}} \right) - \dfrac{{{x^4}}}{{16}}\]

Note:
1) In the given integration we can see that the given integration is a product of polynomials and a logarithmic term hence to find the integration it is necessary that we have to use the integration by part method which is as mentioned in the solution hint.
2) To find the integration by part method we have to take ${x^3}$as the first term and $\ln x$ as the second term and first term is to be integrated and second term is to be differentiated.