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

Apply Integration By Parts

= 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

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