Here, we show you a step-by-step solved example of integrals involving logarithmic functions. This solution was automatically generated by our smart calculator:
We can solve the integral $\int x^2\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Multiplying fractions $\frac{1}{x} \times \frac{x^{3}}{3}$
Multiplying the fraction by $\ln\left|x\right|$
Simplify the fraction $\frac{x^{3}}{3x}$ by $x$
Now replace the values of $u$, $du$ and $v$ in the last formula
Take the constant $\frac{1}{3}$ out of the integral
Multiply the fraction and term in $- \left(\frac{1}{3}\right)\int x^{2}dx$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
Multiplying fractions $-\frac{1}{3} \times \frac{x^{3}}{3}$
The integral $-\int\frac{x^{2}}{3}dx$ results in: $\frac{-x^{3}}{9}$
Gather the results of all integrals
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
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