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

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

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

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

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

Solution steps

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

Expand

\frac{2 x + 3}{2 - 3 x ^{2}} : \frac{2 x}{2 - 3 x ^{2}} + \frac{3}{2 - 3 x ^{2}}

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

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

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

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

Add a constant to the solution

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

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