What is the integral of ((ln(x))^2)/(6x^6) ?

The integral of ((ln(x))^2)/(6x^6) is 1/6 (-(ln^2(x))/(5x^5)+2/5 (-(ln(x))/(5x^5)-1/(25x^5)))+C

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integral of ((ln(x))^2)/(6x^6)

\int \frac{( ln ( x ) ) ^{2}}{6 x ^{6}} d x

\frac{1}{6} (- \frac{ln ^{2} ( x )}{5 x ^{5}} + \frac{2}{5} (- \frac{ln ( x )}{5 x ^{5}} - \frac{1}{25 x ^{5}})) + C

Solution steps

\int \frac{( ln ( x ) ) ^{2}}{6 x ^{6}} d x

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{( ln ( x ) ) ^{2}}{x ^{6}} d x

Apply Integration By Parts

: - \frac{ln ^{2} ( x )}{5 x ^{5}} - \int - \frac{2 ln ( x )}{5 x ^{6}} d x

= \frac{1}{6} (- \frac{ln ^{2} ( x )}{5 x ^{5}} - \int - \frac{2 ln ( x )}{5 x ^{6}} d x)

\int - \frac{2 ln ( x )}{5 x ^{6}} d x = - \frac{2}{5} (- \frac{ln ( x )}{5 x ^{5}} - \frac{1}{25 x ^{5}})

= \frac{1}{6} (- \frac{ln ^{2} ( x )}{5 x ^{5}} - (- \frac{2}{5} (- \frac{ln ( x )}{5 x ^{5}} - \frac{1}{25 x ^{5}})))

Simplify

= \frac{1}{6} (- \frac{ln ^{2} ( x )}{5 x ^{5}} + \frac{2}{5} (- \frac{ln ( x )}{5 x ^{5}} - \frac{1}{25 x ^{5}}))

Add a constant to the solution

= \frac{1}{6} (- \frac{ln ^{2} ( x )}{5 x ^{5}} + \frac{2}{5} (- \frac{ln ( x )}{5 x ^{5}} - \frac{1}{25 x ^{5}})) + C

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What is the integral of ((ln(x))^2)/(6x^6) ?
The integral of ((ln(x))^2)/(6x^6) is 1/6 (-(ln^2(x))/(5x^5)+2/5 (-(ln(x))/(5x^5)-1/(25x^5)))+C