What is the integral of (5-2x)/(sqrt(12x-4x^2)) ?

The integral of (5-2x)/(sqrt(12x-4x^2)) is arcsin(2/3 x-1)+sqrt(x(-x+3))+C

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integral of (5-2x)/(sqrt(12x-4x^2))

\int \frac{5 - 2 x}{12 x - 4 x ^{2}} d x

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

Solution steps

\int \frac{5 - 2 x}{12 x - 4 x ^{2}} d x

Expand

\frac{5 - 2 x}{12 x - 4 x ^{2}} : \frac{5}{2 3 x - x ^{2}} - \frac{x}{3 x - x ^{2}}

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

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

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

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

Simplify

\frac{5}{2} arcsin (\frac{2}{3} x - 1) - (- x (- x + 3) + \frac{3}{2} arcsin (\frac{2}{3} x - 1)) : arcsin (\frac{2}{3} x - 1) + x (- x + 3)

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

Add a constant to the solution

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

2^{x}

\int tan^{3} (2 - x) sec (2 - x) d x

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\int e^{3 x} d x

What is the integral of (5-2x)/(sqrt(12x-4x^2)) ?
The integral of (5-2x)/(sqrt(12x-4x^2)) is arcsin(2/3 x-1)+sqrt(x(-x+3))+C