What is the integral of 2arcsin(2x) ?

The integral of 2arcsin(2x) is 2xarcsin(2x)+sqrt(1-4x^2)+C

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integral of 2arcsin(2x)

\int 2 arcsin (2 x) d x

2 x arcsin (2 x) + 1 - 4 x^{2} + C

Solution steps

\int 2 arcsin (2 x) d x

Take the constant out:

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

= 2 \cdot \int arcsin (2 x) d x

Apply Integration By Parts

= 2 (x arcsin (2 x) - \int \frac{2 x}{1 - 4 x ^{2}} d x)

\int \frac{2 x}{1 - 4 x ^{2}} d x = - \frac{1}{2} 1 - 4 x^{2}

= 2 (x arcsin (2 x) - (- \frac{1}{2} 1 - 4 x^{2}))

Simplify

2 (x arcsin (2 x) - (- \frac{1}{2} 1 - 4 x^{2})) : 2 x arcsin (2 x) + 1 - 4 x^{2}

= 2 x arcsin (2 x) + 1 - 4 x^{2}

Add a constant to the solution

= 2 x arcsin (2 x) + 1 - 4 x^{2} + C

What is the integral of 2arcsin(2x) ?
The integral of 2arcsin(2x) is 2xarcsin(2x)+sqrt(1-4x^2)+C