What is the integral of (2x+5)/(sqrt(9-x^2)) ?

The integral of (2x+5)/(sqrt(9-x^2)) is -2sqrt(9-x^2)+5arcsin(1/3 x)+C

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integral of (2x+5)/(sqrt(9-x^2))

\int \frac{2 x + 5}{9 - x ^{2}} d x

- 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x) + C

Solution steps

\int \frac{2 x + 5}{9 - x ^{2}} d x

Expand

\frac{2 x + 5}{9 - x ^{2}} : \frac{2 x}{9 - x ^{2}} + \frac{5}{9 - x ^{2}}

= \int \frac{2 x}{9 - x ^{2}} + \frac{5}{9 - 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{2 x}{9 - x ^{2}} d x + \int \frac{5}{9 - x ^{2}} d x

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

\int \frac{5}{9 - x ^{2}} d x = 5 arcsin (\frac{1}{3} x)

= - 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x)

Add a constant to the solution

= - 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x) + C

y = sin (e^{2 x})

\frac{d}{d x} (\frac{x + 3}{8 x ^{2} e ^{x}})

x \to 0 + lim (2^{x})

\int \frac{e ^{x}}{e ^{2 x} + e ^{x} + 2} d x

\frac{d}{d x} ((e^{x^{2}} + 4)^{2} (4 x)^{3 x})

What is the integral of (2x+5)/(sqrt(9-x^2)) ?
The integral of (2x+5)/(sqrt(9-x^2)) is -2sqrt(9-x^2)+5arcsin(1/3 x)+C