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

The integral of 1/(sqrt(2-2x-x^2)) is arcsin(1/(sqrt(3))(x+1))+C

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

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

arcsin (\frac{1}{3} (x + 1)) + C

Solution steps

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

Complete the square

2 - 2 x - x^{2} : - (x + 1)^{2} + 3

= \int \frac{1}{- ( x + 1 ) ^{2} + 3} d x

Apply u-substitution

: \int \frac{1}{- u ^{2} + 3} d u

= \int \frac{1}{- u ^{2} + 3} 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}{3} (x + 1))

Simplify

= arcsin (\frac{1}{3} (x + 1))

Add a constant to the solution

= arcsin (\frac{1}{3} (x + 1)) + C

\int (e^{x} + \frac{1}{x}) d x

\int \frac{1}{3 - 3 x ^{2}} d x

\int \frac{2 x}{( x - 3 ) ^{5}} d x

\int x sin (x) e^{x} d x

\int sec^{4} (x) tan^{5} (x) d x

What is the integral of 1/(sqrt(2-2x-x^2)) ?
The integral of 1/(sqrt(2-2x-x^2)) is arcsin(1/(sqrt(3))(x+1))+C