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
ANSWERED
asked 2021-05-23
Find the indefinite integral \(\int \ln(\frac{x}{3})dx\) (a) using a table of integrals and (b) using the Integration by parts method.

Expert Answers (1)

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
\(\int\ln(\frac{x}{3})\times1 dx=\ln(\frac{x}{3})\times\int 1 dx - \int [\frac{d}{dx}[\ln(\frac{x}{3})]\times\int 1 dx]+C\)
\(=\ln(\frac{x}{3})\times x - \int(\frac{1}{\frac{x}{3}}\times\frac{1}{3}\times x)dx+C\)
\(=\ln(\frac{x}{3})\times x-\int 1 dx+C\)
\(=x\ln(\frac{x}{3})-x+C\)

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