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

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

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

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

4 - x^{2} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= - \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

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

Apply the Power Rule

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

Substitute back

u = 4 - x^{2}

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

Simplify

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

= 4 - x^{2}

Add a constant to the solution

= 4 - x^{2} + C

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