Final answer to the problem
Step-by-step Solution
How should I solve this problem?
- Choose an option
- Integrate by partial fractions
- Integrate by substitution
- Integrate by parts
- Integrate using tabular integration
- Integrate by trigonometric substitution
- Weierstrass Substitution
- Integrate using trigonometric identities
- Integrate using basic integrals
- Product of Binomials with Common Term
- Load more...
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 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 $\frac{1}{2}$
Multiplying fractions $\frac{1}{x} \times \frac{2\sqrt{x^{3}}}{3}$
Multiplying the fraction by $\ln\left|x\right|$
Taking the constant ($2$) out of the integral
Simplify the fraction $\frac{\sqrt{x^{3}}}{3x}$ 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 $-2\cdot \left(\frac{1}{3}\right)\int\sqrt{x}dx$
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 $\frac{1}{2}$
Simplify the expression
The integral $-2\int\frac{\sqrt{x}}{3}dx$ results in: $\frac{-4\sqrt{x^{3}}}{9}$
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 $C$
