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

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

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

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

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

Solution steps

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

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

Simplify

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

= - - x^{2} - 4 x

Add a constant to the solution

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

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