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

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

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

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Take the constant out:

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

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

Simplify

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

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

= \frac{1}{ln} (- \frac{ln ^{2} ( x )}{2 x ^{2}} - (\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*ln(x^3))) ?
The integral of (ln(x))/((x^3*ln(x^3))) is 1/(ln)(-(ln^2(x))/(2x^2)-((ln(x))/(2x^2)+1/(4x^2)))+C