What is the integral of x*(-2)/(sqrt(1-4x^2)) ?

The integral of x*(-2)/(sqrt(1-4x^2)) is 1/2 sqrt(1-4x^2)+C

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integral of x*(-2)/(sqrt(1-4x^2))

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

\frac{1}{2} 1 - 4 x^{2} + C

Solution steps

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

Simplify

x \frac{1}{1 - 4 x ^{2}} : \frac{x}{1 - 4 x ^{2}}

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

Apply u-substitution

= - 2 \cdot \int - \frac{1}{8 u} d u

Take the constant out:

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

= - 2 (- \frac{1}{8} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

= - 2 (- \frac{1}{8} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 1 - 4 x^{2}

= - 2 (- \frac{1}{8} \cdot 2 (1 - 4 x^{2})^{\frac{1}{2}})

Simplify

- 2 (- \frac{1}{8} \cdot 2 (1 - 4 x^{2})^{\frac{1}{2}}) : \frac{1}{2} 1 - 4 x^{2}

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

Add a constant to the solution

= \frac{1}{2} 1 - 4 x^{2} + C

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