What is the integral of-(ln(x))/(x^3) ?

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

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integral of-(ln(x))/(x^3)

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Simplify

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

Add a constant to the solution

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

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