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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

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

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

Apply Integration By Parts

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

\int - 1 d x = - x

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

Simplify

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

Add a constant to the solution

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

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