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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

\int t 1 - t d t

\int \frac{6}{x ^{2}} cos (\frac{6}{x}) d x

\int ln (\frac{1}{x} + 1) d x

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

\int \frac{x + 3}{2 x ^{2} + 8 x + 20} 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