What is the integral of (2-x)/(sqrt(9-x^2)) ?

The integral of (2-x)/(sqrt(9-x^2)) is 2arcsin(1/3 x)-xarcsin(1/3 x)+xarcsin(x/3)+sqrt(9-x^2)+C

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integral of (2-x)/(sqrt(9-x^2))

\int \frac{2 - x}{9 - x ^{2}} d x

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

Solution steps

\int \frac{2 - x}{9 - x ^{2}} d x

Apply Integration By Parts

: arcsin (\frac{1}{3} x) (2 - x) - \int - arcsin (\frac{1}{3} x) d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (2-x)/(sqrt(9-x^2)) ?
The integral of (2-x)/(sqrt(9-x^2)) is 2arcsin(1/3 x)-xarcsin(1/3 x)+xarcsin(x/3)+sqrt(9-x^2)+C