Question

# Find the indefinite integral \int \ln(\frac{x}{3})dx (a) using a table of integrals and (b) using the Integration by parts method.

Applications of integrals
Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.

2021-05-24

Given,
$$\int \ln(\frac{x}{3})dx$$
a).
By using the table of integrals, we have $$\int \ln x dx = x\ln (x)-x+C$$
Therefore,
$$\int \ln (\frac{x}{3})dx=\frac{\frac{x}{3}\times\ln(\frac{x}{3})-\frac{x}{3}}{\frac{1}{3}}+C$$ (x-coefficient must be divided)
$$=x\ln(\frac{x}{3})-x+C$$
Step 2
b).
$$\int \ln (\frac{x}{3})dx=\int \ln (\frac{x}{3})\times1dx$$
Now integrating by parts, we get
$$\int\ln(\frac{x}{3})\times1 dx=\ln(\frac{x}{3})\times\int 1 dx - \int [\frac{d}{dx}[\ln(\frac{x}{3})]\times\int 1 dx]+C$$
$$=\ln(\frac{x}{3})\times x - \int(\frac{1}{\frac{x}{3}}\times\frac{1}{3}\times x)dx+C$$
$$=\ln(\frac{x}{3})\times x-\int 1 dx+C$$
$$=x\ln(\frac{x}{3})-x+C$$