What is the integral of (3-5x)/(sqrt(9-5x^2)) ?

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

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integral of (3-5x)/(sqrt(9-5x^2))

\int \frac{3 - 5 x}{9 - 5 x ^{2}} d x

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

Solution steps

\int \frac{3 - 5 x}{9 - 5 x ^{2}} d x

Expand

\frac{3 - 5 x}{9 - 5 x ^{2}} : \frac{3}{9 - 5 x ^{2}} - \frac{5 x}{9 - 5 x ^{2}}

= \int \frac{3}{9 - 5 x ^{2}} - \frac{5 x}{9 - 5 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{3}{9 - 5 x ^{2}} d x - \int \frac{5 x}{9 - 5 x ^{2}} d x

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

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

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

Simplify

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

Add a constant to the solution

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

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