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

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

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

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

- - x^{2} + 2 x + 2 + arcsin (\frac{1}{3} (x - 1)) + C

Solution steps

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

Complete the square

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

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

Apply u-substitution

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

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Substitute back

u = x - 1

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

Expand

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

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

Add a constant to the solution

= - - x^{2} + 2 x + 2 + arcsin (\frac{1}{3} (x - 1)) + C

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