Question

What is the integral of \[\int{x\ln x}\]?

Answer 1

Hint: The given function is a composite function which is a product of two different functions. To solve the given question, we should know the product rule of integration which is used to integrate functions of the form \[f(x)g(x)\]. The product rule states that \[\int{f(x)g(x)}\] is evaluated as \[f(x)\int{g(x)dx}-\int{\left( f'(x)\int{g(x)} \right)dx}\]. For the given question, we will take the functions to be \[f(x)=lnx\And g(x)=x\]. We will use the product rule to integrate the function.

Complete step by step solution:
We are asked to integrate the function \[\int{x\ln x}\]. This function is of the form \[f(x)g(x)\], so we will use the product rule of integration to evaluate its integration. The product rule states that \[\int{f(x)g(x)}\] is evaluated as \[f(x)\int{g(x)dx}-\int{\left( f'(x)\int{g(x)} \right)dx}\]. For the given question, we will take the functions to be \[f(x)=lnx\And g(x)=x\].
We know that the integration of x with respect to x that is \[\int{xdx}\] is \[\dfrac{{{x}^{2}}}{2}\]. The differentiation of lnx with respect to x is \[\dfrac{1}{x}\].
\[\int{f(x)g(x)}=f(x)\int{g(x)dx}-\int{\left( f'(x)\int{g(x)} \right)dx}\]
Putting the function, we get
\[\int{\left( \ln x \right)(x)}=\ln x\int{xdx}-\int{\left( \dfrac{d\left( \ln x \right)}{dx}\int{xdx} \right)dx}\]
We have already evaluated the required integrals and derivatives for above expression, by substituting them we get
\[\Rightarrow \int{\left( \ln x \right)(x)}=\ln x\dfrac{{{x}^{2}}}{2}-\int{\left( \dfrac{1}{x}\times \dfrac{{{x}^{2}}}{2} \right)dx}\]
Further simplifying the above expression, we get
 \[\begin{align}
  & \Rightarrow \int{\left( \ln x \right)(x)}=\ln x\dfrac{{{x}^{2}}}{2}-\int{\dfrac{x}{2}dx} \\
 & \Rightarrow \int{\left( \ln x \right)(x)}=\ln x\dfrac{{{x}^{2}}}{2}-\dfrac{{{x}^{2}}}{4} \\
\end{align}\]
As this is an indefinite integration, we must add the constant of integration. Thus, the final answer is \[\int{\left( \ln x \right)(x)}=\ln x\dfrac{{{x}^{2}}}{2}-\dfrac{{{x}^{2}}}{4}+C\].

Note:In the above example, you may think that what if we take the functions as \[g(x)=\ln x\And f(x)=x\]. As we will need the value of \[\int{g(x)}dx\] in the product rule of the integration we should know the value of \[\int{\ln xdx}\] that’s why we did not use this assumption. The constant of integration is very important, one should not forget to write it.