What is the integral of (2-3x)/(sqrt(3-4x^2)) ?

The integral of (2-3x)/(sqrt(3-4x^2)) is arcsin(2/(sqrt(3))x)+3/4 sqrt(3-4x^2)+C

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integral of (2-3x)/(sqrt(3-4x^2))

\int \frac{2 - 3 x}{3 - 4 x ^{2}} d x

arcsin (\frac{2}{3} x) + \frac{3}{4} 3 - 4 x^{2} + C

Solution steps

\int \frac{2 - 3 x}{3 - 4 x ^{2}} d x

Expand

\frac{2 - 3 x}{3 - 4 x ^{2}} : \frac{2}{3 - 4 x ^{2}} - \frac{3 x}{3 - 4 x ^{2}}

= \int \frac{2}{3 - 4 x ^{2}} - \frac{3 x}{3 - 4 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}{3 - 4 x ^{2}} d x - \int \frac{3 x}{3 - 4 x ^{2}} d x

\int \frac{2}{3 - 4 x ^{2}} d x = arcsin (\frac{2}{3} x)

\int \frac{3 x}{3 - 4 x ^{2}} d x = - \frac{3}{4} 3 - 4 x^{2}

= arcsin (\frac{2}{3} x) - (- \frac{3}{4} 3 - 4 x^{2})

Simplify

= arcsin (\frac{2}{3} x) + \frac{3}{4} 3 - 4 x^{2}

Add a constant to the solution

= arcsin (\frac{2}{3} x) + \frac{3}{4} 3 - 4 x^{2} + C

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What is the integral of (2-3x)/(sqrt(3-4x^2)) ?
The integral of (2-3x)/(sqrt(3-4x^2)) is arcsin(2/(sqrt(3))x)+3/4 sqrt(3-4x^2)+C