What is the integral of ln(x)(sqrt(1-(ln(x))^2))/x ?

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

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integral of ln(x)(sqrt(1-(ln(x))^2))/x

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

- \frac{1}{3} (1 - ln^{2} (x))^{\frac{3}{2}} + C

Solution steps

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

Apply u-substitution

: \int u 1 - u^{2} d u

= \int u 1 - u^{2} d u

Apply u-substitution

: \int - v^{2} d v

= \int - v^{2} d v

Take the constant out:

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

= - \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= - \frac{v ^{3}}{3}

Substitute back

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{3} (1 - ln^{2} (x))^{\frac{3}{2}} + C

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