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

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

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

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

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

Solution steps

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

Complete the square

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

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int 1 d v

= \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 + 1))

Simplify

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

Add a constant to the solution

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

2 x^{3} + 3 x^{2} - 12 x + 7

\frac{d}{d t} (\frac{t ^{3}}{3})

2 x^{2} + x + 1

f (x) = 5 - 4 x^{2}

\frac{d}{d x} (2 x - \frac{2}{x})

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