What is the integral of (-5)/(sqrt(4-2x-x^2)) ?

The integral of (-5)/(sqrt(4-2x-x^2)) is -5arcsin(1/(sqrt(5))(x+1))+C

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integral of (-5)/(sqrt(4-2x-x^2))

\int \frac{- 5}{4 - 2 x - x ^{2}} d x

- 5 arcsin (\frac{1}{5} (x + 1)) + C

Solution steps

\int \frac{- 5}{4 - 2 x - x ^{2}} d x

Take the constant out:

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

= - 5 \cdot \int \frac{1}{4 - 2 x - x ^{2}} d x

Complete the square

4 - 2 x - x^{2} : - (x + 1)^{2} + 5

= - 5 \cdot \int \frac{1}{- ( x + 1 ) ^{2} + 5} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 5} d u

= - 5 \cdot \int \frac{1}{- u ^{2} + 5} d u

Apply Trigonometric Substitution

: \int 1 d v

= - 5 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= - 5 \cdot 1 \cdot v

Substitute back

= - 5 \cdot 1 \cdot arcsin (\frac{1}{5} (x + 1))

Simplify

= - 5 arcsin (\frac{1}{5} (x + 1))

Add a constant to the solution

= - 5 arcsin (\frac{1}{5} (x + 1)) + C

\frac{d}{d x} (\frac{sin ( x )}{x ^{5}})

\frac{d}{d x} (x^{4} - 2 x^{2} + 3)

x^{3} - 2 x + 1

y = (x^{3} - 9 x)^{10}, (3, 0)

y = 3 x^{3} - 3 x, (- 4, 0)

What is the integral of (-5)/(sqrt(4-2x-x^2)) ?
The integral of (-5)/(sqrt(4-2x-x^2)) is -5arcsin(1/(sqrt(5))(x+1))+C