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

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

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

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

arcsin (2 x) + C

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

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

= 2 \cdot \frac{1}{2} arcsin (u)

Substitute back

u = 2 x

= 2 \cdot \frac{1}{2} arcsin (2 x)

Simplify

2 \cdot \frac{1}{2} arcsin (2 x) : arcsin (2 x)

= arcsin (2 x)

Add a constant to the solution

= arcsin (2 x) + C

y = 5 x^{8} e^{x}

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

1 - 8 x^{2} y^{^{'}} = x

\int sin (x) + c d x

\frac{1}{( s + 1 ) ( ( s + 2 ) ^{2} + 2 )}

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