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

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

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

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

- 2 x^{2} - 4 + C

Solution steps

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

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

u = x^{2} - 4

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

Simplify

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

= - 2 x^{2} - 4

Add a constant to the solution

= - 2 x^{2} - 4 + C

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