What is the integral of (4ln(x))/(x^3) ?

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

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integral of (4ln(x))/(x^3)

\int \frac{4 ln ( x )}{x ^{3}} d x

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

Solution steps

\int \frac{4 ln ( x )}{x ^{3}} d x

Take the constant out:

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

= 4 \cdot \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

= 4 (- \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}}

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

Simplify

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

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

Add a constant to the solution

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

f (x) = \frac{c ^{4} - x ^{4}}{c ^{4} + x ^{4}}

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