Question

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

Answer 1

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

& \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{aligned}$$ As this is an indefinite integration, we must add the constant of integration. Thus, the final answer is $$\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.