Question

# Evaluate the integral. (Use C for the constant of integration.) \int \ln (\sqrt x)dx

Integrals
Evaluate the integral. (Use C for the constant of integration.)
$$\int \ln (\sqrt x)dx$$

2021-05-03
2021-09-09

Solution:

$$\int\ln(\sqrt{x})dx=\int1\cdot\ln(\sqrt{x})dx$$

Now we apply integration by parts

$$=[x\cdot\ln(\sqrt{x})]-\int\frac{1}{\sqrt{x}}(\frac{1}{2\sqrt{x}})\cdot xdx+C$$

$$=[x\ln(\sqrt{x})]-\int\frac{1}{2x}\cdot xdx+C$$

$$=x\ln\sqrt{x}-\frac{1}{2}\int dx+C$$

$$\int\ln(\sqrt{x})dx=x\ln\sqrt{x}-\frac{x}{2}+C$$