What is the integral of x/(sqrt(8+2x-x^2)) ?

The integral of x/(sqrt(8+2x-x^2)) is -sqrt(-x^2+2x+8)+arcsin(1/3 (x-1))+C

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integral of x/(sqrt(8+2x-x^2))

\int \frac{x}{8 + 2 x - x ^{2}} d x

- - x^{2} + 2 x + 8 + arcsin (\frac{1}{3} (x - 1)) + C

Solution steps

\int \frac{x}{8 + 2 x - x ^{2}} d x

Complete the square

8 + 2 x - x^{2} : - (x - 1)^{2} + 9

= \int \frac{x}{- ( x - 1 ) ^{2} + 9} d x

Apply u-substitution

= \int \frac{u + 1}{- u ^{2} + 9} d u

Expand

\frac{u + 1}{- u ^{2} + 9} : \frac{u}{- u ^{2} + 9} + \frac{1}{- u ^{2} + 9}

= \int \frac{u}{- u ^{2} + 9} + \frac{1}{- u ^{2} + 9} 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} + 9} d u + \int \frac{1}{- u ^{2} + 9} d u

\int \frac{u}{- u ^{2} + 9} d u = - - u^{2} + 9

\int \frac{1}{- u ^{2} + 9} d u = arcsin (\frac{1}{3} u)

= - - u^{2} + 9 + arcsin (\frac{1}{3} u)

Substitute back

u = x - 1

= - - (x - 1)^{2} + 9 + arcsin (\frac{1}{3} (x - 1))

Expand

- (x - 1)^{2} + 9 : - x^{2} + 2 x + 8

= - - x^{2} + 2 x + 8 + arcsin (\frac{1}{3} (x - 1))

Add a constant to the solution

= - - x^{2} + 2 x + 8 + arcsin (\frac{1}{3} (x - 1)) + C

What is the integral of x/(sqrt(8+2x-x^2)) ?
The integral of x/(sqrt(8+2x-x^2)) is -sqrt(-x^2+2x+8)+arcsin(1/3 (x-1))+C