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

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

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

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

- \frac{x}{arcsin ( x )} + Ci (arcsin (x)) + C

Solution steps

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

Apply u-substitution

: \int \frac{sin ( u )}{u ^{2}} d u

= \int \frac{sin ( u )}{u ^{2}} d u

Apply Integration By Parts

: - \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u

= - \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u

\int - \frac{cos ( u )}{u} d u = - Ci (u)

= - \frac{sin ( u )}{u} - (- Ci (u))

Substitute back

u = arcsin (x)

= - \frac{sin ( arcsin ( x ) )}{arcsin ( x )} - (- Ci (arcsin (x)))

Simplify

- \frac{sin ( arcsin ( x ) )}{arcsin ( x )} - (- Ci (arcsin (x))) : - \frac{x}{arcsin ( x )} + Ci (arcsin (x))

= - \frac{x}{arcsin ( x )} + Ci (arcsin (x))

Add a constant to the solution

= - \frac{x}{arcsin ( x )} + Ci (arcsin (x)) + C

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