What is the integral of (ln(x^5))/(5x^3) ?

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

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integral of (ln(x^5))/(5x^3)

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

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

Solution steps

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

Take the constant out:

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

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

Simplify

ln (x^{5}) : 5 ln (x)

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

Take the constant out:

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

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

Apply Integration By Parts

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

Simplify

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

= - \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

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