What is the integral of 1/2 x^{-3}ln(3/5 x) ?

The integral of 1/2 x^{-3}ln(3/5 x) is 1/2 (-(ln(3/5 x))/(2x^2)-1/(4x^2))+C

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integral of 1/2 x^{-3}ln(3/5 x)

\int \frac{1}{2} x^{- 3} ln (\frac{3}{5} x) d x

\frac{1}{2} (- \frac{ln ( \frac{3}{5} x )}{2 x ^{2}} - \frac{1}{4 x ^{2}}) + C

Solution steps

\int \frac{1}{2} x^{- 3} ln (\frac{3}{5} x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x^{- 3} ln (\frac{3}{5} x) d x

Apply Integration By Parts

: - \frac{ln ( \frac{3}{5} x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x

= \frac{1}{2} (- \frac{ln ( \frac{3}{5} x )}{2 x ^{2}} - \int - \frac{1}{2 x ^{3}} d x)

\int - \frac{1}{2 x ^{3}} d x = \frac{1}{4 x ^{2}}

= \frac{1}{2} (- \frac{ln ( \frac{3}{5} x )}{2 x ^{2}} - \frac{1}{4 x ^{2}})

Add a constant to the solution

= \frac{1}{2} (- \frac{ln ( \frac{3}{5} x )}{2 x ^{2}} - \frac{1}{4 x ^{2}}) + C

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What is the integral of 1/2 x^{-3}ln(3/5 x) ?
The integral of 1/2 x^{-3}ln(3/5 x) is 1/2 (-(ln(3/5 x))/(2x^2)-1/(4x^2))+C