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

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

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integral of ((1-x))/(sqrt(8+2x-x^2))

\int \frac{( 1 - x )}{8 + 2 x - x ^{2}} d x

8 + 2 x - x^{2} + C

Solution steps

\int \frac{1 - x}{8 + 2 x - x ^{2}} d x

Apply u-substitution

: \int \frac{1}{2 u} d u

= \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 8 + 2 x - x^{2}

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

Simplify

\frac{1}{2} \cdot 2 (8 + 2 x - x^{2})^{\frac{1}{2}} : 8 + 2 x - x^{2}

= 8 + 2 x - x^{2}

Add a constant to the solution

= 8 + 2 x - x^{2} + C

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