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

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

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

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

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

Solution steps

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

Simplify

x^{- 3} ln (5 x) : \frac{ln ( 5 x )}{x ^{3}}

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

Apply Integration By Parts

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

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

Add a constant to the solution

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

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\frac{\partial}{\partial z} (z)

\frac{3}{( x - 1 )}

\int \frac{e ^{x} - 1}{e ^{2 x}} d x

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