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

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

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

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

- arcsin (x + 1) - - x^{2} - 2 x + C

Solution steps

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

Apply Integration By Parts

: x arcsin (x + 1) - \int arcsin (x + 1) d x

= x arcsin (x + 1) - \int arcsin (x + 1) d x

\int arcsin (x + 1) d x = arcsin (x + 1) (x + 1) + - x^{2} - 2 x

= x arcsin (x + 1) - (arcsin (x + 1) (x + 1) + - x^{2} - 2 x)

Simplify

x arcsin (x + 1) - (arcsin (x + 1) (x + 1) + - x^{2} - 2 x) : - arcsin (x + 1) - - x^{2} - 2 x

= - arcsin (x + 1) - - x^{2} - 2 x

Add a constant to the solution

= - arcsin (x + 1) - - x^{2} - 2 x + C

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