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

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

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

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int 3 sin (u) + 2 d u

= \int 3 sin (u) + 2 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 3 sin (u) d u + \int 2 d u

\int 3 sin (u) d u = - 3 cos (u)

\int 2 d u = 2 u

= - 3 cos (u) + 2 u

Substitute back

u = arcsin (\frac{1}{3} x)

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

Simplify

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

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

Add a constant to the solution

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

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