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

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

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

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

arcsin (\frac{1}{2} x + 1) + C

Solution steps

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

Complete the square

- x^{2} - 4 x : - (x + 2)^{2} + 4

= \int \frac{1}{- ( x + 2 ) ^{2} + 4} d x

Apply u-substitution

= \int \frac{1}{- u ^{2} + 4} d u

Apply Trigonometric Substitution

= \int 1 d v

Integral of a constant:

\int a d x = a x

= 1 \cdot v

Substitute back

= 1 \cdot arcsin (\frac{1}{2} (x + 2))

Simplify

1 \cdot arcsin (\frac{1}{2} (x + 2)) : arcsin (\frac{1}{2} x + 1)

= arcsin (\frac{1}{2} x + 1)

Add a constant to the solution

= arcsin (\frac{1}{2} x + 1) + C

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