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

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

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integral of (3+x)/(sqrt(9-x^2))

\int \frac{3 + x}{9 - x ^{2}} d x

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

Solution steps

\int \frac{3 + x}{9 - x ^{2}} d x

Apply Trigonometric Substitution

= \int 3 (sin (u) + 1) d u

Take the constant out:

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

= 3 \cdot \int sin (u) + 1 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

= 3 (\int sin (u) d u + \int 1 d u)

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

\int 1 d u = u

= 3 (- cos (u) + u)

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

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