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 $\int x\ln\left(4x\right)dx$ 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 $\displaystyle f'(x)=\frac{a'}{a}$
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
Multiplying the fraction by $4$
First, identify or choose $u$ and calculate it's derivative, $du$
Now, identify $dv$ and calculate $v$
Solve the integral to find $v$
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Multiplying fractions $\frac{1}{x} \times \frac{1}{2}$
Multiplying the fraction by $x^2$
Simplify the fraction $\frac{x^2}{2x}$ by $x$
Now replace the values of $u$, $du$ and $v$ in the last formula
Take the constant $\frac{1}{2}$ out of the integral
Multiply the fraction and term in $- \left(\frac{1}{2}\right)\int xdx$
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
Multiplying fractions $-\frac{1}{2} \times \frac{1}{2}$
The integral $-\int\frac{x}{2}dx$ results in: $-\frac{1}{4}x^2$
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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