Final Answer
Step-by-step Solution
Problem to solve:
Specify the solving method
We can solve the integral $\int\sqrt{x}\ln\left(x\right)dx$ 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 $u$ and calculate $du$
Now, identify $dv$ and calculate $v$
Solve the integral
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 $\frac{1}{2}$
The integral of a function times a constant ($\frac{2}{3}$) is equal to the constant times the integral of the function
Multiplying the fraction by $\sqrt{x^{3}}$
Simplify the fraction $\frac{\sqrt{x^{3}}}{x}$ by $x$
Now replace the values of $u$, $du$ and $v$ in the last formula
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 $\frac{1}{2}$
Simplify the expression inside the integral
The integral $-\frac{2}{3}\int\sqrt{x}dx$ results in: $-\frac{4}{9}\sqrt{x^{3}}$
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 $C$