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

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

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integral of x/(sqrt(3+2x-x^2))

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

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

Solution steps

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

Complete the square

3 + 2 x - x^{2} : - (x - 1)^{2} + 4

= \int \frac{x}{- ( x - 1 ) ^{2} + 4} d x

Apply u-substitution

: \int \frac{u + 1}{- u ^{2} + 4} d u

= \int \frac{u + 1}{- u ^{2} + 4} d u

Expand

\frac{u + 1}{- u ^{2} + 4} : \frac{u}{- u ^{2} + 4} + \frac{1}{- u ^{2} + 4}

= \int \frac{u}{- u ^{2} + 4} + \frac{1}{- u ^{2} + 4} 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} + 4} d u + \int \frac{1}{- u ^{2} + 4} d u

\int \frac{u}{- u ^{2} + 4} d u = - - u^{2} + 4

\int \frac{1}{- u ^{2} + 4} d u = arcsin (\frac{1}{2} u)

= - - u^{2} + 4 + arcsin (\frac{1}{2} u)

Substitute back

u = x - 1

= - - (x - 1)^{2} + 4 + arcsin (\frac{1}{2} (x - 1))

Expand

- (x - 1)^{2} + 4 : - x^{2} + 2 x + 3

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

Add a constant to the solution

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

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