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

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

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

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

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

Solution steps

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

Expand

\frac{x + 2}{3 + 2 x - x ^{2}} : \frac{x}{3 + 2 x - x ^{2}} + \frac{2}{3 + 2 x - x ^{2}}

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

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

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

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

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

Simplify

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

Add a constant to the solution

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

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