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

rocedwrp

rocedwrp

Answered question

2021-05-23

Find the indefinite integral ln(x3)dx (a) using a table of integrals and (b) using the Integration by parts method.

Answer & Explanation

tafzijdeq

tafzijdeq

Skilled2021-05-24Added 92 answers

Given,
ln(x3)dx
a).
By using the table of integrals, we have lnxdx=xln(x)x+C
Therefore,
ln(x3)dx=x3×ln(x3)x313+C (x-coefficient must be divided)
=xln(x3)x+C
Step 2
b).
ln(x3)dx=ln(x3)×1dx
Now integrating by parts, we get
ln(x3)×1dx=ln(x3)×1dx[ddx[ln(x3)]×1dx]+C
=ln(x3)×x(1x3×13×x)dx+C
=ln(x3)×x1dx+C
=xln(x3)x+C

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