Integral of $\sqrt{x} \ln\left(x\right)$

Find $\int \sqrt{x} \ln\left(x\right)\, dx$.

For the integral $\int{\sqrt{x} \ln{\left(x \right)} d x}$, use integration by parts $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$.

Let $\operatorname{u}=\ln{\left(x \right)}$ and $\operatorname{dv}=\sqrt{x} dx$.

Then $\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$ (steps can be seen ») and $\operatorname{v}=\int{\sqrt{x} d x}=\frac{2 x^{\frac{3}{2}}}{3}$ (steps can be seen »).

The integral becomes

$${\color{red}{\int{\sqrt{x} \ln{\left(x \right)} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \frac{2 x^{\frac{3}{2}}}{3}-\int{\frac{2 x^{\frac{3}{2}}}{3} \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \int{\frac{2 \sqrt{x}}{3} d x}\right)}}$$

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=\frac{2}{3}$ and $f{\left(x \right)} = \sqrt{x}$:

$$\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - {\color{red}{\int{\frac{2 \sqrt{x}}{3} d x}}} = \frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - {\color{red}{\left(\frac{2 \int{\sqrt{x} d x}}{3}\right)}}$$

Apply the power rule $\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$ $\left(n \neq -1 \right)$ with $n=\frac{1}{2}$:

$$\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\int{\sqrt{x} d x}}}}{3}=\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\int{x^{\frac{1}{2}} d x}}}}{3}=\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}}}}{3}=\frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \frac{2 {\color{red}{\left(\frac{2 x^{\frac{3}{2}}}{3}\right)}}}{3}$$

Therefore,

$$\int{\sqrt{x} \ln{\left(x \right)} d x} = \frac{2 x^{\frac{3}{2}} \ln{\left(x \right)}}{3} - \frac{4 x^{\frac{3}{2}}}{9}$$

Simplify:

$$\int{\sqrt{x} \ln{\left(x \right)} d x} = \frac{2 x^{\frac{3}{2}} \left(3 \ln{\left(x \right)} - 2\right)}{9}$$

Add the constant of integration:

$$\int{\sqrt{x} \ln{\left(x \right)} d x} = \frac{2 x^{\frac{3}{2}} \left(3 \ln{\left(x \right)} - 2\right)}{9}+C$$

$\int \sqrt{x} \ln\left(x\right)\, dx = \frac{2 x^{\frac{3}{2}} \left(3 \ln\left(x\right) - 2\right)}{9} + C$A