What is the integral of (arcsin(ln(x)))/x ?

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

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integral of (arcsin(ln(x)))/x

\int \frac{arcsin ( ln ( x ) )}{x} d x

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

Solution steps

\int \frac{arcsin ( ln ( x ) )}{x} d x

Apply u-substitution

: \int arcsin (u) d u

= \int arcsin (u) d u

Apply Integration By Parts

: u arcsin (u) - \int \frac{u}{1 - u ^{2}} d u

= u arcsin (u) - \int \frac{u}{1 - u ^{2}} d u

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

= u arcsin (u) - (- 1 - u^{2})

Substitute back

u = ln (x)

= ln (x) arcsin (ln (x)) - (- 1 - ln^{2} (x))

Simplify

= ln (x) arcsin (ln (x)) + 1 - ln^{2} (x)

Add a constant to the solution

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

\frac{1}{cos ( \frac{x}{2} )}

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

(x - 5) (x + 4) (x + 6)

\frac{d}{d x} (\frac{( x + 2 ) ( x + 1 )}{2 x - 1})

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